How to delete an icloud account without a password?!

My mother deleted her old apple ID and email connected to it over a year ago. Yet she still cant delete the icloud account using the old apple ID without the now nonexistent password. Her new apple ID was no problem to use in itunes just the icloud. She didnt know it was going to cause such a problem! She cant use her ipad without an interrupting pop up coming up to ask her for her old apple ID password every 5 fricken minutes. This is ridiculous ive tried every possible thing apple care suggests, plus everything i can find on google. Without deleteing everything on my moms ipad, shed loose a ton of stuff from her old account like hundreds of paid for books, which i know would really upset her as it would anyone. Although even after deleting everything i wonder if the damn icloud thing would still stand. Anyways if there is anyone who thinks they can help me id very much appreciate it! You would be the best person in the world!

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