Use iTunes to do a reset to factory default condition. Always be sure to do your backups before fiddling with your iPad. You might do both the iCloud and the iTunes backup, just to be sure!
Back up and Restore your iOS Device with iCloud or iTunes
iTunes: About iOS Backups
Use iTunes to Restore your iOS Device to Factory Settings
Various issues from many different users have been solved by this. I've lost count on how many have been helped by this, so far. But, it doesn't mean EVERYONE will be helped, but I've seen a whole lot. You can only try and see ...
For all iPad 2 owners, this is an important solution for you! These problems result from long-term use, continuous upgrades, corrupted files and various glitches that happen over this long term use. You just need to get your iPad 2 back to FACTORY FRESH CONDITION to start out clean!
Here's one person's solution which will help a lot of iPad 2 users!
SOLVED: IOS 8 Issues on iPad 2:
I just returned home from a visit to the Genius Bar. Like you and many others, when I upgraded my iPad 2 to IOS 8, it became practically unusable. It was VERY slow, lagged whenever trying to to do anything. Lots of screen freezes and apps spontaneously crashing. I was convinced that the processor on the iPad 2 just couldn't handle the new IOS and I went in to see about restoring to IOS 7.
Thankfully, I was hooked up with a very sharp guy at the bar. The problem is not with the OS, it is with it being an *upgraded* OS. He used the analogy of when you had an older Windows computer and how over time (and over upgrades), they just tended to get slower and slower because of all of the garbage that was carried forward in the upgrades. At those times (and if you've ever had a Windows computer, you get this), the only thing that would speed it up would be to wipe the machine and do a fresh install of the current OS and then reload your apps, and the viola, everything would be fast again. Well, it turns out that IOS can suffer the same problems.
THE FIX: FIRST, be sure you have a current iCloud backup of your device. Connect your iPad to a computer that has a clean fresh copy of IOS 8 on iTunes (this is best done with a computer other than the one that you sync your device to because you DO NOT want to restore it from a back up just yet). Ideally, if you do this at an Apple store you'll be assured of the right version there and someone to answer any questions you might have along the way. You then hold the power and home button down to hard-reset the device and continue to hold them down until it reboots into recovery mode. You then follow the instructions on the computer to reinstall the OS on the device (which wipes everything from it and then reinstalls a clean copy of it). Once this is done (which only takes about 15 minutes), your device will reboot and ask you to set it up. Answer the questions about language, location, etc. and continue until it asks you to log into your iTunes account. Once you do that, it will ask if you want to either 1) Set it up as a new device, 2) Restore from iCloud, or 3) Restore from iTunes. Choose (2), restore from iCloud. This will ensure that it brings down "clean" copies of all of your apps and preferences rather than the potentially corrupted ones that might reside in your iTunes backup. That's it! It will take a few hours depending on how much data and apps you have, but I am happy to report that my iPad 2 is now running IOS 8 and it's running as fast as it ever did when it was brand new!
To sum it up, this process 1) Wipes your device and the garbage causing your problems, 2) Loads a clean and bug-free version of IOS 8 to your device, and 3) restores your apps and data so that it looks just like it did before the wipe, with the exception that it works now! :-)
Note also that by following this process you don't have to go through the painful and laborious process of reloading all of your apps, data and/or settings. iCloud takes care of all of that for you.