I understand that you're experiencing problems reaching the Login Screen after patching the game client. This can be caused by a failed patching attempt or other corruption and there are a few ways to fix it.
First, you can try executing the Repair feature in our Game Patcher--if you can load the Game Patcher. There should be a Cog icon on the corner which will lead you to a Settings Menu; once you're there, you can execute a repair.
Second--if the first step didn't work or you cannot launch the Game Patcher--you can try the following to force a re-patch:
1) Go to your Riot Games folder (C:/Riot Games/League of Legends)
2) Open the following directory:
C:\Riot Games\League of Legends\RADS\projects\lol_air_client\releases\<hig hest number folder>
3) Delete the following files:
As always, if these steps do not resolve your issue, please feel free to submit a support ticket through our support site. You can reach it by clicking the SUPPORT link on the top navigation menu or by going straight to http://support.leagueoflegends.com/ We'll be able to help you with your support ticket; please be sure to tell us which steps you've already taken to try to resolve your issue so we can help you quickly.