I managed to get mine fixed by opening up the phone. Turns out, it was just the battery cable that came loose... It could also be the motherboard though, but mine was just the battery cable...
I suggest taking it to Apple to get them to open it up for you, or if you want to do it yourself, use patience and a really tiny screw-driver and a suction cup to pry the screen up. I decided that my phone was already screwed, it couldn't get any more screwed up anyway, so just opened it, and after reconnecting the loose cable, it works perfect again, so yay.