Ipod Touch 4th Generation won't turn on...Battery problem?

A friend of mine dropped my 4th generation Ipod Touch about a week ago, and for a few days it would shut down anywhere from a few seconds to a few minutes after turning on. It would also occasionally give me a warning that the iPod was too heated up and so I would have to wait a few minutes before using it. I tried to restore it but that didn't help. Then, it stopped turning on all together, except that it says that it is at the lowest battery if you plug it into the computer. It won't show up in iTunes though. I took it to the Genius bar, and they said that since I had previously exchanged the back and front at a 3rd party place, they weren't able to evaluate the problem. Should I send it to a 3rd party place in the hope that getting a new battery wil solve the problem, or should I just give up on it? Thanks so much!!!