Q: problems with apple tv staying connected to iPad

First, try a system reset of the iPad.  It cures many ills and it's quick, easy and harmless...

Hold down the on/off switch and the Home button simultaneously until you see the Apple logo.  Ignore the "Slide to power off" text if it appears.  You will not lose any apps, data, music, movies, settings, etc.

If the Reset doesn't work, try a Restore.  Note that it's nowhere near as quick as a Reset.  It could take well over an hour!  Connect via cable to the computer that you use for sync.  From iTunes, select the iPad/iPod and then select the Summary tab.  Follow the on-screen directions for Restore and be sure to say "yes" to the backup.  You will be warned that all data (apps, music, movies, etc.) will be erased but, as the Restore finishes, you will be asked if you wish the contents of the backup to be copied to the iPad/iPod.  Again, say "yes."

At the end of the basic Restore, you will be asked if you wish to sync the iPad/iPod.  As before, say "yes."  Note that that sync selection will disappear and the Restore will end if you do not respond within a reasonable time.  If that happens, only the apps that are part of the IOS will appear on your device.  Corrective action is simple -  choose manual "Sync" from the bottom right of iTunes.

If you're unable to do the Restore (or it doesn't help), go into Recovery Mode per the instructions here.  You WILL lose all of your data (game scores, etc,) but, for the most part, you can redownload apps and music without being charged again.  Also, if you have IOS-7, read this.