Here is my own modified backup/restore procedure that has already worked, successfully, for quite a few iPad 2/3 users, so far!
THIS PROCEDURE DOES NOT RETURN YOUR iPAD TO ITS PREVIOUS iOS VERSION.
Also, as a disclaimer.
I make no guarantees, expressed or implied, that any of my own procedures will return EVERY older iPad to a usable, running condition.
Make sure you have the latest iTunes version installed, first.
iTunes is up to version is up to 12.3.3 something.
You need, at least OS X 10.8.5 Mountain Lion to install it.
Then connect your iPad to your computer running the latest iTunes and see if your iPad will sync and backup.
Disconnect your iPad from the computer next.
From just the iPad, in the Settings App under the General setting, look in the right column for Reset option, then erase and reset your iPad to completely erase your iPad and then, again, from the iPad, set it up as a new iPad setup to factory "out of the box" operation.
Do this completely from your iPad using your local WiFi network to completely setup your iPad 2.
Then reconnect your iPad to the computer and make sure to cancel the syncing, by tapping the small X to the left in the top center iTunes display box.
iTunes should detect your iPad. If iTunes begins to try and sync, again, cancel the sync.
Select Restore from Backup instead and pick your last backup to restore back to your iPad.
Once the restore is complete, finish setting up the iPad and let it sync back to iTunes you can either cancel the sync, again or let it backup, again.
Then disconnect the iPad.
Do one hard reset of your iPad by holding down both the Home and sleep/wake buttons simultaneously until your iPad goes to black and restarts with Apple logo, then release the buttons.
Once your iPad is back to the Home screen,
In the Settings App under the General tab, to the right under Spotlight sesrch, try disabling the search under apps that really do not need a search, like some games, remotes, apps that really do not need to be searched, etc., to reduce the list for Spotlight to search.
In Settings app, under the Safari settings panel, tap the panel and in the right hand window, turn off/disable Safari Suggestions.
Try turning on Reduced Motion.
This is found in the Settings App in the General tab the left panel.
In the right panel look under Accessibility, the look for Reduce Motion and turn this feature "On".
You should see an appreciable performance increase on all iPad 2, 3 and 4 models.
If you have the iOS Mail app set to automatically fetch mail at one of the default fetch time intervals, in the Settings App, under the Mail, Contacts, Calendar settings panel, tap this panel, in the righthand window, turn off/disable Push Mail and set fetching mail for either every hour or set for manual fetch only for all of your mail accounts.
In the Settings App under the General Tab, in the right column, look for Background App Refresh and turn this setting to "off".
If you purchased any music from iTunes, you will need to redownload that content back to your iPad for playing directly from your iPad and not from Apple Music servers.
Also, if your iPad contained a lot of images in the Photos app, these will take time to regenerate the original AND thumbnail images themselves inside of the Photos app when you first launch it.
Good Luck!
