If just your back light works, but the image doesn't show, hold the power button and bottom volume button simultaneously, this will do a soft reset and fix that problem. The back light and LCD picture goes through the same cable (the cable up near the top left). The bottom two connections for the screen deal with the digitizer only.
Monkeybiten, try the soft reset as well, but you may need to do a hard reset (Google LG G2 hard reset, it will delete everything, so I suggest backing up your info to the cloud). It's quite possible you have a virus (malicious app) that's sucking your phone's resources. You could try an anti-virus. AVG from Google play store is good.