Backlight works ... some times

I purchased a broken 4th generation ipod. The seller listed it as "broken back light" and "turns off when removed from charger."

I was really surprised because when I plugged it into the charger everything seemed to work. The back light even worked. It showed a charge logo. It was fully functional.

I was suspicious. I waited two days, then charged the ipod again. At this point, the back light went off and stayed off. The unit got very hot from charging, and it tells me to "connect to power" when I try to turn it on, despite the fact that it was indeed charged.

So I know I have a broken battery. My question: Is the back light malfunctioning as a result of the heat produced by the broken battery? Would it fix itself if I replace the battery, and not the lcd? Or is this some other type of problem all together?