I had to set up an iPhone 5 and Apple TV via a Telstra 4G WiFi AP for a conference demo and all was working perfectly when the mirroring option abruptly disappeared.
I didn't think to reset networking, which may well have worked, but apart from a a full reset, we tried pretty much everything else.
It looked like we'd have to fall back to our HDMI cable (which had also been tested and worked perfectly). My colleague at the conference tried the cable, but it didn't work for some reason so she thought she'd try AirPlay again on the off-chance and, lo-and-behold, the mirroring option suddenly reappeared.
Perhaps the code that manages mirroring for both HDMI and AirPlay connections is buggy, rather than AirPlay itself and trying the HDMI connection unstuck it somehow.
If you've got an HDMI cable/adaptor for your device, it might save you a full reset.