This photo shows the back of a Kindle Touch. You may want to open a Kindle so you can change the battery yourself in minutes rather than waiting days for your Kindle to come back from a repair station. I had to open this one after it became water-logged in an accident. Fortunately, the electronics survived without damage.
Power the Kindle off. Use a fingernail to pull the back away from the Kindle just a little in the area of the lower half of the Kindle. I cannot say all models of the Kindle will open the same way, but I expect they might.
Step 1: Use a Coin
Insert a thin coin between the back and the Kindle. Twist a little to release the catches on both sides and across the bottom.
Step 2: Removing the Back
Slide the back toward the bottom of the Kindle to release the catches on the upper portion of the back. See the red arrow. The yellow lines show the area where the slide catches are located. See the second photo. The orange lines mark the upper slide catches. The yellow lines show the lower catches.
The process of attaching the back again is the reverse of removing it. Arch the back as shown in the photo from step 2. Secure the upper catches by sliding the back upward (Ignore the red arrow.). This sounds easy, but it is easy for one catch to remain unsecured. See the third photo.
Press the lower half of the back into place until the catches snap and are secure.
Step 3: Replacing the Battery?
Remove the five screws marked with red arrows. There are no connectors to detach, although that can vary by model. Simply lift the battery out of the Kindle and drop the new battery into place. Connector tabs align and make the connection. Two of the screws hold the connectors in contact.
Information about the battery is on the label. New batteries are available here.
When my Kindle got wet, water flowed out as I removed the back. The Kindle was on stand by. It would have been my preference to have had the Kindle powered down completely. I did not have precision screwdrivers and it was a few hours before I was able to remove the battery. I used a hair dryer on "low" to dry the Kindle. I should have let the Kindle air dry a couple of days. When I powered it up after only a day, it did not properly respond to commands, even though it had worked well the evening before, which was after I had used the hair dryer. I decided some remaining water had migrated into the control switches. Everything was fine after I removed the battery again and let the Kindle air dry for a couple of days.