I had this issue as well, SO FRUSTRATING!! To fix it, all I did was find the original launcher, and launch "lol.launcher.admin" instead. You might have to launch it as an administrator, even. I replaced my shortcuts with shortcuts to the .admin launcher, and it works fine.
It still complains about another launcher running, but it properly closes the "other" one and launches the game now.
I remember the installer gave me the option to install the game just for my own account or for all users, and I chose all users. I don't know if this has anything to do with it.
I hope this helps someone.