i think there are 2 possible problems
a faulty battery or a faulty logic board
if the board failed - you'r out of luck, logic boards costs the same as a used nano 4g (sometimes even more)
if the battery failed - replace it. i don't know if you have a mulitmeter to measure the voltage of the battery - but this would be my first test, if the voltage would be lower than 3v - it would be gone, some ipods are simply "dead" when the battery dies - so it's hard to tell if the board died or the battery.