This problem appears to vary somewhat from user to user. However, this worked for me for iPad2 and Apple TV:
1) Confirm you have actually installed iOS8.2 under "About" in Settings on the iPad2.
2) Check under Apple TV settings that you are running version 7.1 of the Apple TV software.
3) Check you have Airplay turned on in the Apple TV settings
4) Reboot your iPad.
5) Swipe up at the bottom of the screen to reveal the notification centre.
6) You should see the Airplay icon, bottom centre of the notification centre window. If so, tap on it.
7) Select Apple TV and confirm you have a blue checkmark on the right hand side.
8) Swipe up gently to reveal the mirroring toggle switch and slide that to the ON position.
9) So long as your TV is tuned to HDMI 2 and your Apple TV is connected to the TV, you should now
see your iPad desktop on the TV again.
Hope that helps at least some fellow users?!