Hi there, I thought I'd share my thoughts on this as well.
I just purchased an iPad Air and could not get the Airplay to work at all. Even when I tried to toggle the"Mirroring" slider it just wouldn't work, or stay on. In the end, I reset everything. I unplugged my TV, Modem, Router (Netgear for what it's worth), Apple TV (2nd Gen), and turned the iPad Air completely off and back on.
I left everything off for about two minutes, turned it all back on, and everything works no problem now. Not sure this will help anyone but it's what worked for me!
My main reason for getting Airplay to work is so I can use the NFL Game Pass and stream it to my HDTV. So now I'm quite happy!
Love the community, thanks for the help! Good luck!