Can't Sign Out of iCloud--Catch 22

My wife was getting security messages on her iPad wanting her to verify her identity. Her apple ID email became inactive months ago, and when trying to verify on the apple ID web site, it was wanting to send an email verification to the invalid email address.

I decided to change the email address of her apple ID to fix the issue. The iTunes store happily accepted the new ID on her iPad, but it would not allow her to sign out of iCloud because her old ID had been changed. A search for others with this issue gave two solutions: upgrade to IOS 9.2, but she is already in 9.3.2 or temporarily change the email address back to the old email and then sign out--it promised that she wouldn't have to verify the old email address.

Unfortunately, changing the email of her Apple ID to the old email address sent verification code to the old email address which is no longer valid.

I even tried resetting her iPad, but it is asking for the password of the old ID which doesn't exist.

What can I do?