ICloud – Problem signing into Apple ID

Perhaps you could try addressing the problem from the other side of the issue. When I faced a similar problem with my now defunct student email address on another account, the solution turned out to be contacting the IT department of my old school and convincing them to temporarily re-instate my student email address. Once I convinced them to do it, it turned out to be a really simple process. I was able to change over and recover passwords from several services tied to the old email address, and once completed just had them deactivate it again. Your milage may vary depending on your educational institution, but it's definitely worth a shot.

One new feature of iOS 7 is an activation lock - and this means that you can't sign out or erase the device without entering the proper password for the account to which your device has been locked.

• http://support.apple.com/kb/HT5818

Towards the bottom of the article, it has a link to use My Apple ID web page to check and reset your password if you have forgotten it.

• https://appleid.apple.com/cgi-bin/WebObjects/MyAppleId.woa/

Presumably, if you can't reset the password, you might need to reach out to Apple directly and see if there is an administrative process such as mailing them proof of purchase or correctly answering questions about past billing events to have them reset the account password.