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Hello from JustAnswer.
No Neurologist has yet responded, however, with my experience as a Medical Director, I have several comments about the letter.
I would vastly simplify the letter, apologizing for previous behavior, asking for return to the practice, and making a brief statement that you are better from a psychiatric perspective, but instead of going into detail, asking for the Neurologist to discuss your case with your current Psychiatrist. It would be more persuasive for the Neurologist to hear of your improvement from your Psychiatrist, rather than from you.
I also would recommend that you do not enclose a check for any amount of money. It would be fine to include a statement that you would cover any expenses involved in returning to the practice in advance, but the Neurologist can let you know of the expense.
If I can provide any additional information.