This solution did not work for me. I still get a black TV screen until I close X-plane on my Iphone 6 plus. Then the mirroring works fine again. In fact, I use the mirroring with several other flight simulators. Mirroring works great with "Aerofly 2", and on "MetalStorm" the airplay integrated dualscreen mode works like a dream.
Well, I finally got X-plane to mirror by restarting Apple TV 3 times.