Hello Sandclimber,
I'm sorry to hear you are having issues with your Apple TV and your iPad Air. If you are having difficulties connecting via peer-to-peer AirPlay, you may want to double check that Bluetooth is enabled on both devices, as noted in the following article:
With peer-to-peer AirPlay, your iOS device and Apple TV don't need to be on the same Wi-Fi network. If you have one of these products, you can use peer-to-peer AirPlay:
* iPhone 5 or later, iPad (4th generation or later), iPad mini, or iPod touch (5th generation) using iOS 8 or later.
* Apple TV (3rd generation rev A, model A1469 or later) with Apple TV software 7.0 or later. Find out which Apple TV model you have.
To use peer-to-peer AirPlay, your iOS device needs to have both Wi-Fi and Bluetooth turned on.
You'll need an Internet connection to stream iTunes content.
Use AirPlay to wirelessly stream content from your iPhone, iPad, or iPod touch - Apple Support
Sincerely,
- Brenden