Katarina is mia cuz she went to get a rework expect her to be back by tuesday. 

logic at its finest

casseiopia she  hates her sister

The kidnappers are Cassiopeia and the might of Demacia himself....Garen.

Motive: Cassiopeia is out for revenge, her duty to her house is what eventually lead to her transformation from beautiful temptress to hideous monstrosity. Garen is out for justice, with one strike he has been delivered his greatest rival and dealt a massive financial blow to one of Noxus's most powerful houses.

How: The kidnapper demanded a route that took Cassiopeia near the Demacian border, which leads to a Demacian co-conspirator being involved. Cassiopeia took the money to Garen who then sapped her to add credibility to their ruse. The story Cassiopeia told is obviously false do to not being able to identify a cardigan from the back. Garen planted a cardigan near the scene to tie the story together and escaped detection by hiding in a near-by bush.

Twitch is not involved due to not fitting the "culprits" description, hes not a tall man, he is a short rat. Graves is most likely not involved because his behavior is to be expected given his priors. He is a cunning con-man, and would not lend himself to be caught so easily so close to the crime scene, more likely a case of wrong place/wrong time.

Er, then why did she even join the League in the first place?

The solution to this brainteaser couldn't be any easier. When searching the vicinity for the culprit, the kidnapper was already miles away from the crime scene.

The kidnapper, Twisted Fate the card master, is known to have tricked many people out of their money, never leaving as much as a trace behind.

The cardigan Cassiopeia noted was just misdirection, as to be expected from a professional magician. Even though one cannot differ a cardigan from a regular sweater from behind, he tricked her into believing a cardigan was what she saw, all in order to cast suspicion on Cassiopeia.

Unexpectedly, however, his accomplice was no other than lovely Katarina, the victim herself. Fed up with her father's avarice, she decided to increase her funds for the upcoming shopping tour at Noxus' most notorious leather goods store. 

On a side note, the involvement of Twisted Fate also explains the presence of Graves in the vicinity of the crime. As we all know, Graves is on an eternal quest to take out his revenge on Twisted Fate for conning him in the past. After many years, he had finally tracked down the card master; the knife he was carrying was, in fact, not meant for self-protection, but to finally satisfy his bloodthirst and fulfill his nemesis' destiny.

Destiny, however, didn't mean for this to happen. It was destiny itself that allowed Twisted Fate to forsee the upcoming  assassination and flee to a remote place.

With that, two of the mysteries of this case are solved. The presence of a plague rat cannot be explained rationally, however rumour has it that Twitch, who has recently undergone excessive training and changed his job from an assassin lurking through dark forests towards heavy artillery, was trying to adopt a few useful tricks from flavour-of-the-month heavy-hitter Graves.

The only person who could've done it was Cassiopeia. You can't tell if someone is wearing a Cardigan when you only see the back of them.

You would see this from the front.

http://www.shortlike.com/wp-content/uploads/2012/06/cardigans-on-men.jpg

And you would see a long sweater from the back. Cassiopeia lied in her testimony and for what reason? Of course to direct suspicion to other people. Cassiopeia lied and kidnapped Katarina.

Um, what the hell is a cardigan?  I saw a picture of a man wearing a Mr. Roger's sweater.

Next, she said front, so she could of seen the sides or an angle of the sides.  Since she lives at court, she probably knows much about fashion.  All she would have to do is match what she saw with what she has seen previously in court.

Internal Noxian politics. Jericho Swain commissioned Graves( a man with no affiliation) with information about Twisted Fate. Swain offered Cassiopeia a chance to make her sister lose face and therefore improve her own standing in the Family. Cassiopeia lures Katarina into a vulnerable position for her to be taken by Graves. The ransom is 1) a means to measure the wealth of the De'conteau family 2) a means to Make the General seem weak to his supporters and therefore strengthen Swain's own position for Noxian High Commander.

Cassiopeia and Twitch were the kidnappers, Cassiopeia was looking for money and the homeless Twitch wouldn't turn down the offer from Cass to kidnap Kat and get a cut of the ransom. Cassiopeia needed someone to hit her over the head, because if she had done it herself, or slammed her head against something, the angles would be wrong. It also allows her to give some information to cop, such as the cardigan, and twitch would have the alibi saying, "I just found this, I'm poor, I don't know what you're talking about." Implying the actual thief dumped the cardigan whilst retreating.

I've actually seen this one a long time ago.. Cass is the liar and thief... you cant tell if a sweater is a cardigan from the back as it buttons up the front.

To start off, you told us that the culprit was male "a lone kidnapper telephoned the mansion and made his demands." 

Next we know that only the kidnapper could have known where Cassiopeia was going because of her secretiveness during her travels. 

We also know that the culprit was tall and was wearing a cardigan, is what cass had said. We also know that the culprit can run quickly and also sneak very well.

These rule out a lot of champions such as yordles and not very finesse characters (aka mundo). 

One final clue is that the character has to be someone that the Du Couteau family could not have known, someone from a far away place. Such as ionia. I belive that shen is the culprit, but not without a little bit of help. 

The backstory on this is: Shen traveled from ionia to help with the doctors in demacia. During his travels he saw that Du Couteau was making a lot of money from others despair. Because of Shen's need to create balance he was urged to kidnap katarina, possibly by taunting, then drugging her. He then ordered a ransom to be taken to dimacia. But he got some help, he knew that the prisoner graves had been released recently, as some of his patients were shoved around or knew about him, and that is why they were in the hospital. He figured that graves could redeem himself for the bad that he had done if he helped shen restore balance, and also would split some of the ransom. Shen knew when he had to catch Cassiopeia, but was running late trying to save a persons life. So he through on his cardigan over his doctors uniform and set out to find cass. Shen hit her with a ki strike, ran around the corner, threw out his cardigan and tele'd to graves using stand united. who was far enough away at the time for shen to taunt to safety. He also gave graves one of his knifes to keep him protected. Later twitch found the cardigan (which had been eaten by the moths that already follow twitch, nothing lasts long in that rats possesion). And that is who captured katarina and stole the money from cass. 

The answer is Shen and Graves

The kidknapper has to be Yorick.

Reason 1: Twitch found a moth-eaten cardigan and Yorick being a zombie, naturally wears shaggy clothes.

Reason 2: A shovel can very easily be used to attack Casseopeia from behind+ Yorick's omen of war gives him a speed buff so he can quickly run away after hitting Cass.

Reason 3: The lamp. Cass said she saw the back of the criminal in the light of a lamp. Yorick has a lamp protruding from his body that hangs over his head at all times.

I think it was Jax because...
1. She saw him next to a LAMP POST instead of a streetlight, she was just delusional when she was hit.
2. He is fast. She heard footsteps and then a quick Leapstrike to the head for a knockout.

3. He's tall and has a dark cardigan.

The answer can only be one person: Caitlyn.

It can not be Graves nor Twitch, as neither of them would be able to knock Cass unconscious. Graves gun simply isn't meant for beating people with it, and would likely break. Twitch's crossbow would surely break if it were used as a bludgeon. However, Caitlyn, on the other hand, has her rifle which looks similar to a World War 1 or 2 rifle, which were designed so that the back could be used in close quarters combat. This would easily be able to knock out Cass from behind.

Caitlyn is also the one able to catch Katarina, since she can Shunpo out of Twitch's or Graves's hands since neither has a stun or snare. Caitlyn has her Traps that would be able to catch her and not let her escape. 

Since Caitlyn has knowledge of the city, she knew that Graves and Twitch would be in the area, so she purposefully set traps in areas they usually walk so that there would be two prime suspects.

All this evidence points to the one and only culprit, CAITLYN.

The answer is simple really. A smelly stinking rat.

From the observations we can clearly deduce the answer. First, how did Katarina, a famous champion in the League of Legends be kidnapped? Even Garen, the might of Demacia, is afraid of 'the sinister blade'. (that's why he hides in brush so often)

It could only be done to trickery...Perhaps through an 'ambush'

Now, the next thing is how Cassiopeia brought the gold but on her way did not see anyone. Yet when she came to the meeting place, she heard footsteps and got smacked on the head. Since stealth does not hide the sound of your footsteps, and after hitting someone stealth disappears...

Now the last question, where did the gold go? Remember how Graves and Twitch were both empty handed? Remember how twitch was chilling on the dump? He was not looking for food or friends, no my good man, he was hiding the gold! Nobody is going to dig for gold in a dump, so it fits perfectly.

 PS: Also, Twitch mentioned a 'sweater' he was not wearing. Why would you do that (if not asked for) unless you know people saw you with it while committing the crime?