Airplay not connecting

I work at a school. We have about 30 Apple TVs in different classrooms connected wirelessly to Meraki APs. Some of our instructors are having issues using AirPlay Mirroring from their iOS devices. I should mention that in these classrooms, there are about 30 students each using an iPad while the instructor is trying to connect to the ATV. We have our ATVs configured with a passcode. The instructor will get the prompt for the passcode, they enter it in to their device like normal, but then nothing happens.The device gets the blue Airplay title bar, but the ATV sits on the main screen indefinitely. In this state other users can no longer connect to the ATV because "Someone else is using the device." We've followed the AirPlay troubleshooting document(http://support.apple.com/kb/TS4215), with no luck. Also checked for interference. All devices have a great connection.

Anyone have any ideas as to what's wrong?

I should add that they're 3rd Gen ATVs running 6.0.2 and iPad Airs running 7.0.4.