MR member Strether (karma coming your way!), kindly shared a YouTube video he found on how to change the battery in the Kindle 2. It is crazy simple.
If you try this, you will void your warranty and there is no guarantee that Amazon (as generous as they usually are) will help you if you mess it up! Try this at your own risk. (save the old battery so you can put it back in if you need to 
)
That said, my Kindle 2 is out of warranty with Amazon but I purchased the extended warranty so I have another year to go. I would still give this a try. The company
Newpower99 is selling the batteries for $24.95.
Enjoy the video and try to ignore the hokey "Bolero" music in the background.