I like to tinker with electronics and give a new life
Impossible. Try opening using a ifixit Essential Toolkit, and measuring the dimensions of the battery using Vernier Calipers. Use those dimensions to search for a lipo battery on eBay. Mostly you will find one (atleast $15-20). Use a multimeter in Voltage mode to test the leads of battery. Desolder them and carefully solder battery (keep in mind that the leads are very thick). In that case, short the leads to length, use emery wire to extend and then solder to the board. Then you can bend the Backcover to shape and reseat it. I would use some mobile adhesive like B-7000 to keep it shut. Totally depends on your skill level as this is very very hard and you must know micro-soldering to achieve this. If any other part gets damaged, find a donar iPod to exchange parts. LiPo battery usually has a limited lifetime. It costs you lot of time, and I would attempt this, if I'm the only survivor on the Earth after Doomsday.