I'm assuming you tried a hard reset (press menu and center button together for 5sec)and a firmware restore through itnes. If hard resetting and restoringit doesn't work and the Apple store doesn't help you, you have to replacethe logic board. If it was the battery only, it would have to turn on when plugged into the charger. There is a service called ipod repair nyc that can do the replacement for you can use an iPod repair guide that shows you how to replace the screen yourself, the part is on ebay.
More than likely your battery is dead. Replacement will surely help. Talk to an ipod reseller for details on authorised repairers
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