iPod Nano 1st Gen Battery Replacement Woe

PROBLEM: After my iPod Nano Model A1137 battery replacement, reset, and when plugged into USB AC power, the iPod reported “Charged” and works PERFECTLY! However unplugged, the controls never work, the screen fades out in a few seconds, and then it’s dead. Back on AC, the logo displays briefly, battery icon shows fully charged, and in seconds it works PERFECTLY! When plugged into USB dock instead, the screen flashes Apple logo on/off, never charges, and does not mount in iTunes. I have reset the iPod multiple times, have followed <http://www.apple.com/support/ipod/five_rs/nano/#>, I but have not yet closed the back cover.

HISTORY: For my iPod Nano Model A1137, I purchased "Replacement battery for ipod Nano 1st gen a1137...” from "why-not-save” via eBay. But the new battery had the wire colors red-BLUE-black, instead of my existing red-WHITE-black. The vendor suggested I simply swap the WHITE for BLUE. Before I soldered it in, I checked the voltage across all wires and found it matched the polarity of the old battery. I used a needle-point soldering iron, careful not to overheat the board. I only removed the cover; I did not take out the logic board.