This might not be the popular answer but you contracted with this mom to provide care for her child until either give notice because it not suit you. Transporting daughter to school when you are contracted to working is not okay when you have contractual obligations. I do understand that our family come first but I don't agree with running personal errand on business time. You opt put your child in school knowing that she need transportation. You either have to let client know you not available to provide care when you committed to or find someone else take her. I would be cross if my provider suddenly not able provide service they agreed because they choose do something different in business hours.
That said - you did not commit to collect her child from school either. If she opted to put her child in outside program for part of his normal time with you, it up to her to arrange his transportation. What if you not have car insured for business use? She would have to figure out then.
Personally, I think the combination of your change and her change mean the service agreed not suit either of you. Contract and service and times need be changed and if no agreement made then give notice and she find someone who can pick up her child.