Download the amazon video app to an iphone. When watching the video, find the familiar Airplay button somewhere on the screen and direct the video to the Apple TV as usual.
http://www.businessinsider.com/how-to-watch-amazon-videos-on-apple-tv-2016-3/#all-youll-need-isthe-amazon-video-app-an-iphone-or-ipad-and-of-course-an-apple-tv-1
