I think the mom has put the cart before the horse here. She doesn't understand how the job market works. The daughter must apply through the normal channels - which it sounds like she did. If she gets an interview, she needs to stand on her own 2 feet there. At some point, it will come up in the interview or on an application she is asked to fill out, "How did you hear about our company?" or "References". She can list your name at that point. If someone in the hiring area (HR, or the department being served by interns) wants to ask you about her, then you tell the truth. She was your children's babysitter on a number of occasions and seemed very polite and respectful. Period. You're not in a position to evaluate her professionally because you have not seen her in that setting. She should have other references, including college professors and prior summer jobs. Whether her family is nice or not is irrelevant - if you even bring that up, it will make you sound personally involved rather than professional.
Do not say that the internship will be a great experience for the young woman. The company is not in the business of helping out others - their primary goal is to get free or low cost help, and possibly to pre-screen some permanent employees. Your focus must absolutely be on the company's needs, not the girl's. That would apply even if your situation were not tenuous, but it's even more important because you have some bumps in your own road.
You are putting the cart before the horse as well, in that you are worrying about where she will be assigned or what she will think of the company. If they offer her an internship, they will already know that you and she are acquainted. It's up to HR or the hiring manager or the department supervisor to make the matches. If she is placed with you, your job is to be entirely professional, as if you never knew her before.
In any case, you keep her mom out of it. 100%. If you need to, go back and tell the mom only that you must follow hiring protocols but that, if asked, you can give her daughter a personal reference. Do not discuss your work problems or the company's deficiencies. Do not share anything at all about the office, either its high points or low points. If asked, say that you must be entirely professional in this situation, and that the daughter needs to work directly with the HR staff, for everyone's well-being.
It might be better if you went to the daughter directly, but it's interesting to me that she hasn't come to you herself. Maybe she is following hiring protocol herself, and the mom is just getting too involved.