You can get Kindle Replacement Kits for your model of Kindle on eBay for reasonable prices. If you can't get it working, you haven't lost too much. The battery in my Kindle was at the state it lasted about 30 minutes after charging.
Above are pictures I took during working on my older Kindle. No idea where the glue might be in different models.
There are YouTube videos on doing this using heat guns to loosen the glue. That seems risky to me. I used Isopropyl Alcohol as it dissolves glue and is a general electronics cleaning product. I usually have a litre on hand for paint/ink and other general cleanup.
The Kindle is not designed to be repaired. The large pad of glue on the battery cover to the back panel, makes undoing the clips really difficult, and using lots of guitar plectrums helped to keep unclipped tabs from closing again. There are a lot of clips in the back panel, and it has to bend quite a bit to get it off, but the glue pad stops it from bending. The butter knife and IPA into that area of the rear cover loosens the grip of that glue. Eventually.
In my unit the whole underside of the battery was attached with glue, Filling that battery trough with just enough IPA, and working the glue with the butter knife got it out, but it is difficult. You don't want to damage the old battery in anyway. They are a fire hazard.
Once I had the swap done and cover back on, it took some 10 to 20 minutes for the battery to charge enough and the unit to boot, so I knew it was working again. Until then it looks dead.
Don't think this is a task for someone that doesn't have any electronics experience.