I'm having a similar issue.
A couple days ago my computer forced me to do a system restore, since then the LoL Client will not open after selecting the Launch button from the screen that loads after the badge. I've had this happen to me before but usually repeatedly clicking the launcher and telling it to launch until it does has worked before. But now it just disappears and when I go into my task manager it shows the client is running but the log in screen never pops up. I have tried everything listed in this thread to repair the issue. http://forums.na.leagueoflegends.com/board/showthread.php?t=784361
The only ways that I can get the client to launch is to go into my C:>Riot Games>League of Legends and click lol.launcher.admin and then it sometimes lets me into the log in screen after clicking launch. If that doesn't work I have to open the launcher (doesn't matter whether as admin or not I have found) then go to C:>Riot Games>League of Legends>RADS>projects>lol_air_client>releases>0.0.1.131>deploy> then right click LolClient.exe and run as admin and the log in screen pops right up and I'm able to log in.
I'm running Win 7.
You should try right clicking the launcher and selecting run as admin first to see if that works though.