why won't my ipad connect to apple tv after updating software

Welcome to the Apple Community.

Try the following steps, check whether things are working after each step where appropriate, before trying the next.

1. Check AirPlay is turned on on the Apple TV (turn it off and on if it already is)
2. Check that both devices are on the same network (Settings > Wifi, on the mobile device and Settings > General > Network, on the Apple TV).
3. Restart the Apple TV (Settings > General > Restart).
4. Restart the Apple TV by removing ALL the cables for 30 seconds.
5. Restart your router.
6. Restart your mobile device.

I had the same problem and solved it by making sure apple tv was turned on before attempting to find the airplay apple tv icon on my devices. Before the update the airplay icon for apple tv showed even if apple tv was powered down.