I am having the same problem. I flushed DNS and reconfig'd as suggested, and it still doesn't work.
I followed a remedy posted in a different thread of deleting certain folders within the League of Legends folder and subsequent re-patch, and while that worked yesterday, upon closing of the client and restart I am back to my launcher not starting the client. I should not have to re-patch for 20-30 minutes before playing the game every time, and I should not have to leave my LoL client open for days to work around this (inb4 QQ).
on a related topic: Riot needs to skip releasing the next character and spend an ENTIRE 2 weeks of development and manpower to fix bugs and client related issues (the forums have plenty). Ever since Dreamhack there has been a marked decrease in game quality, server up-time, and customer service initiatives.