I had the same problem (type in correct login info, name grays out, password disappears, login sound, but loginscreen music continues playing and nothing happens). I went to my LeagueOfLegends/RADS and deleted all text documents and files (not directories/folders). I deleted everything in my LeagueOfLegends/RADS/temp folder. Deleted everything my LeagueOfLegends/Rads/downloads folder. Essentially went around deleting text documents and small files that are not core to the game. Upon relaunching league, I had to press Accept 8 times, did not have to download anything new. Click Play, then logged in as normal. The game seemed to launch faster, but otherwise back to normal.