How do I change the Apple ID on an iPad when the associated email no longer exists?

Do you mean how do you change the one on the iPad, or do you mean how to change you account email address using the iPad?

On the iPad: It should change itself if you have changed it elsewhere. Try restarting the iPad.

If you mean change your Apple ID , the support article Change your Apple Account primary email address - Apple Support outlines various methods for changing the email address (Apple ID) you use for your Apple Account while retaining everything in your account.

- Note a recent change to the article where you can now change the email address without having to have access to your old email account.

So, you changed the email address used on the same account for a different address on a different device? Can both devices connect to the Internet? If I change my email address used for my account on an iPad and I go to an iPhone, it should show the change there too as long at they can both connect to the Internet and iCloud. It may take a bit of time so wait a while.