The Airplay icon is rather obscure on IOS devices.
1. Once you are past your passcode screen then
2. Double clcik on your iphone button (the one towards the bottom of your phone)
3. This will bring up all of your "running apps"
4. Swipe ALL the way to the left and you will see your volume control bar
5. The Airplay icon is right beside the volume control bar
6. Select the Airplayicon and select Apple TV and delesect iPhone
- this will allow your device to use your Apple TV as a mirroring device.
Yes that is very odd.... here is a useful link from Apple support
It is sounding like it might be a network problem when I read that link. I'm running my stuff through Apple Airport.
Best of luck, I suspect you are very close to a solution.