The iPad 2 is the oldest and least powerful device that can still be upgraded to iOS 9.2. Many users have had success running iOS 9.2 and some have had issues with it on their iPad 2's. I would start will all of the basic things and then if necessary, back up, restore to factory settings and restore the backup or restore as new as a last resort.
Clear Safari. Go to Settings>Safari>Clear History and Website Data
Close all apps. Press the Home button two times quickly. You'll see small previews of your recently used apps. Swipe left to find the app you want to close. Swipe up on the app's preview to close it
Force restart the device. To force restart your device, press and hold both the Sleep/Wake and Home buttons for at least ten seconds, until you see the Apple logo.
If that doesn't help, try resetting all settings. Settings>General>Reset>Reset All Settings. No data is lost when you do this, but all device settings have to be entered again, including your WiFi network passwords.
Restore to factory settings using iTunes
http://support.apple.com/kb/ht1414
If you restore as new, past purchases can be downloaded again.
http://support.apple.com/kb/ht2519