Why won't my iPod hold a charge or sync?

My iPod worked perfectly for years...then I dropped it. It broke in half at the seam. I squished it back together and it worked fine. This happened about 4 times. Then I dropped it again. It split in half and this time a little piece of gray plastic fell out. I shoved it where I thought it should go and closed it back up. Only this time it didn't work. The words and picture on the screen were broken up and the blacklight dimmmed then went out. I thought it was gone for good until I got home and plugged it into my wall charger and it booted back up like nothing was wrong. But when I unplugged it everything died. When I plug into the computer it says "very low battery. please wait." But never charges or syncs.

Is the battery just blown or is it a more serious problem?