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.