You can still update apps manually by opening the Play Store tapping the three vertical lines at the top left and choosing My apps & games, then the Updates tab. Now go back to the main Play Store page and tap the three vertical lines at the top left, then scroll down to choose Settings and tap Auto-update apps > Do not auto-update apps. Open the Play Store app, find the Kindle app page in it, and choose Uninstall.
Glitch: E-books disappearing or not working Since not everyone has the most recent Fire tablet, here are a few common issues on the older range of tablets. You can also set it to turn on automatically.
Hold down the power button until the Power off box pops up and then tap OK.
Or you could factory reset the device and install apps selectively, watching out for any changes in battery drain. You could go to Settings > Apps & Games > Manage All Applications and try uninstalling apps one by one and testing to see if you can find the culprit.