If you really want to do it yourself, you’ll have to crack it open (use a hobby knife and cut/pry it open along the seam), then figure out the battery size (it’s a Lithium Polymer but idk the mA/hr rating or dimensions). Then find a source of LiPo batteries, you can try allbattery.com
A new battery is probably in the $10 - $15 range, maybe less if you get lucky and there’s a blowout sale… You might try searching the ABI (Anything But Ipod) forum for others who may have done this already…
Be careful with the knife because if you jab the battery accidentally, it can erupt and spew fire and goo all over the place.
After desoldering the old battery and replacing it with the new one, you’ll need to glue the case back together. You Clip willl be uglier but at least it should work again.