You may need to connect to a Wi-Fi network to set it up initially…not sure.
Also have you tried to slide your finger up from the bottom of your device (may have to do it a few times) to bring up your control center?
You should see the Airplay icon listed there.
I never tried to connect my son's iPad mini with my ATV3 until just now. I just checked his iPad Mini and did what I just told you to do and when I clicked on Airplay in the control center. The name of my Apple TV popped up on the screen and I could mirror whatever content was on the iPad Mini could now be seen on the Apple TV.
Make sure you have Bluetooth turned ON with your iPad Air.
On your ATV make sure Bluetooth is turned on with that too.
In the settings for the ATV go down to Airplay and make sure that is turned on.
For Security I have mine checked as "None" (meaning anyone on your WiFi network can use Airplay with your ATV. I believe there are a couple other options if you don't want your guests accessing your ATV.
I have Play from iTunes in the Cloud set to Auto (ATV will play content from iTunes in the Cloud instead of your Airplay device when possible. You can also turn this option off which will just play content from an Airplay device only.
See if any of that helps.
Chuck