I just bought a used touch from a local craigslist-equivalent and similar issue - no power on at all. It sort of worked when it was connected to a charger, but would die instantly when power was removed. Then I noticed the case was popped open a bit near the top.
Removing the back revealed a burst LiPo battery
The seller of course claimed ignorance, saying that she'd "never used it."
I soldered an old lithium cell phone battery I had laying around in and it's back to life. $25 for a replacement OEM cell from the 'net. Maybe once this one dies.