Sorry to say but if you didn't document the note you'll have to give them another 2 weeks notice or risk them hounding you for the two weeks payment. They could file in small claims court and you'd have to prove that you gave them 2 weeks notice.
Did you hand the note to someone? If so, name that person and that would be documentation of your giving them notice. Did you talk with someone about the notice in specific terms? If so that is documentation. Did you keep a dated copy. That's documentation.
What did the teacher specifically say to you? If she said she knew you were leaving that is documentation. But if she just said something vague then it's not proof that she knew.
You may have to accept this as a learning experience. Always keep a dated copy of notes, not just ones that give notice. And, always talk specifically to someone about leaving or about making changes or about requesting information and make note of the conversation on a calendar or in a journal.
If you decide to leave then start building documentation as best as you can from memory in preparation of a small claims suit. Write down conversations and approximate dates not only about leaving but about your efforts to gain co-operation from the facility. That way, if you go to small claims you have the ability to show that you worked up to leaving a note to end services. That might help your case.