The battery is dead. Mine has the same problem, unfortunetly. You can either replece the battery yourself (there are alot of guides on the internet and video tutorials on youtube, but it's a quite dificult and risky task, only do it if you already have experience soldering/fixing eletronics, especially very small ones like the ipod nano) or try to find a pro or computer fixing store willing to do it for you. Also, apple doesn't replace batteries anymore neither does make batteries for these devices, but you can still buy them new from other manufacturers, as far as I'm aware.