The Amazon Kindle App for the iPhone and iPad has been a battery hog for the last month. Even when you aren’t reading an e-book and doing something else, the app remains in standby mode.
The Kindle app was updated April 15th 2015 with the quintessential “performance and stability improvements”. This last update was the culprit and many people are finding their tablets or smartphones are not lasting as long as they should. I have an iPhone 6 Plus and most of the time I get about a day and a half of usage. The Kindle app has relegated it to about 15 hours, before it runs out of juice.
I think the main problem is primarily due to a Whispersync timing issue. This is the Amazon system that fetches new content or books you purchased on other devices. There is no way to disable this with Background App Refresh, because Amazon does not have that feature on iOS 8.
Amazon is aware of this issue and is working on a fix. Likely it will be pushed out in about a week.