The Apple ID exists on Apple's servers. Even if you delete it from the iPad, it still exists. Whatever problems you are having you should, eventually, be able to get the iPad to reconnect to it.
The first thing's to try are Restarting and/or Reseting the iPad. This should help with any persistent popups, at least long enough for you to get things back to normal. I'll paste the instructions for that below.
Once you've got access to the Settings app, go to iCloud settings. If there is no account signed in, sign in with your wife's Apple ID. If there is, try signing out, and then sign back in with her Apple ID.
You should also go to the the iTunes and App Store setting to do the same.
Good luck.
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The Usual Fixes
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When you have problems with an app it's a good idea to clear it from RAM.
Close/Clear the app:
iOS 6: With the problem app closed, double tap on the Home button to get the multitask bar. Tap and hold on any icon in the bar until they jiggle, then tap the minus sign on the problem app.
iOS 7: Double tap the Home button to get the multitasking display. Swipe up on the problem app's thumbnail image (not the icon).
If that does not work try a restart,
Restart: Hold down the power button until the power off slider appears. Slide it off. When the iPad finishes shutting down, hold down the power button again until the Apple logo appears. Apple recommends you restart the iPad at least once a month.
and then a reset if that does not work.
Reset: Hold down the power and home buttons until the Apple logo appears. The iPad will reboot. This usually takes a bit longer than a restart.
Reset All Settings: Another, somewhat more drastic thing you can do is Reset All Settings. You'll find this under Settings > General > Reset > Reset All Settings. It won't erase any of your content, apps, or accounts; but you will have to go through most of your other settings and re-tweak them back to your liking.
Do not confuse this with Erase All Content and Settings.
More detailed information at:
http://support.apple.com/kb/TS3281?viewlocale=en_US&locale=en_US