OK - one last suggestion...a reboot.
Turn the iPad off.
Now press and hold the Power and Home buttons (for about 9 seconds) until the white Apple logo appears. The iPad will reboot.
No apps or other user data are lost. Probably won't make any difference, but it's worth a try.
If that fails, then I suspect you must have a hardware fault. Is this a new iPad? If it's still under warranty, I'd return it to the Apple Store, if there's one close to you.
Tim