Sounds like you have a dead battery. You can find a replacement battery [http://www.directfix.com/product/IP-2684.html|here] (''caveat emptor'' - I don't know if this is a reputable vendor or not) and a teardown guide [https://www.ifixit.com/Store/iPod/iPod-Nano-7th-Gen|here]. There is also a Youtube video [https://www.youtube.com/watch?v=_uLzivmjk3Y|here].
Take a look at these resources and see if this is something you want to try yourself.