There are different fixes for different problems. My problem occurred while trying to open league while connected to wireless but behind a hotspot login gate. I got a UserKernel popup and then the launcher never started again.
I still had the .dmg from install, and the fix (after mounting the .dmg) was to copy the UserKernel.app directory from the dmg to the installation dir.
$ mv /Applications/League\ of\ Legends.app/\ Contents/LOL/RADS/system/UserKernel.app \ /Applications/League\ of\ Legends.app/\ Contents/LOL/RADS/system/UserKernel.app.bak $ cp -R /Volumes/League\ of\ Legends\ NA/League\ of\ Legends.app/\ Contents/LOL/RADS/system/UserKernel.app \ /Applications/League\ of\ Legends.app/\ Contents/LOL/RADS/system/
I sent this off to relevant people, but wanted to post a fix here in case it helps people searching google and avoiding forum posts because of all the noise.