It is possible to replace the battery, but usually not feasible to do so.
A kit with the battery, tools, and instructions to do it yourself is about $40. To have a tech do it will run you about double that.
You can buy a brand new kindle for as little as $50: http://amzn.to/2iJrngN
When you receive your replacement kindle, once it is registered and connected to wifi, all of your existing books/purchases will sync to the new kindle, so you won’t lose any of the content you’ve previously purchased.
If you'd prefer to change the battery though I can help you with parts/repair options.
Regarding the personal documents, if you go into the Manage Your Content and Devices:
function within your amazon account (on your computer), you can pick the kindle then set up an email address for it (new or old) and anything you send to that address will be loaded to your kindle. You may of course continue to manually transfer documents and files to the new or old kindle via usb as well.
If you have any other questions, just let me know. Otherwise, please take a moment to rate my service as this is the only way I am credited for my time. Hopefully I have earned 5 Stars but if not please let me know what I can do to ensure you are satisfied. I also invite you to contact me directly with any future support needs at http://bit.ly/1QwLxGh and ask for Nathan.
Thank you, ***** ***** a nice day,