Try to run it as administrator. If that doesn't work:
Go into your League of Legends folder in C:/Riot Games
Open up the "RADS" folder.
Open up the "Projects" folder.
Open up the "lol_air_client" folder.
Open up the "releases" folder.
Open up the folder that's in there. It changes every patch, but contains the same stuff.
Delete the two files named "S_OK" and "release manifest"
Open up the "deploy" folder.
Delete the "logs", "META-INF", "lol.properties", and "LoLClient.exe"
Restart your Patcher.
It will repatch the recent patch you had just downloaded.
This will usually fix it. If not I suggest trying to do a repair from the patcher's options. It takes from 10-20 minutes.