Putting on my business professor hat again.... I would not use the wording "I am no longer willing to provide care." If she shows the email to others, it looks as though you are being petty. I have heard the background and think you are absolutely justified in terminating, but I would still be careful how I word this. I would simply say, "I regret to inform you that I am unable to continue to provide care for (child's name) effective September 12th. I wish you all the best in finding care for (child's name) that best meets the needs of your family. Sincerely......."
I would never put it in writing that we have difficulties, or accuse someone of not wanting to discuss/resolve issues. Why make her any more angry? It does not benefit you in any way. Your goal is to be done with them to avoid future issues, not to lash out and let her have it. That would be seen as unprofessional (as much as I sympathize with your frustration!).