I found this online site based in China, that could get a hold of the parts for the ipod nano 6th generation.
They have a weird set up, but they have the parts or will get a hold of parts for you.
Here's the link:
Hope this helps, because all the searching help me. Good luck with that
This site has the battery on stock:
We don't carry batteries for them yet, but that does not mean you can't find them with a little Google searching. Unfortunately, the hard part is replacing the battery. The 6th Gen Nano is not the easiest to take apart and the battery is soldered to the logic board.