I would have went with the lady to take us both to the school and ensured my child got to school safely, then used the school phone, to call a tow truck to have the car towed.
Then I would thank the lady for her trouble. If I had to work, I would call in with the situation.
BUT I smell BS with this lady if she is late to school on a regular basis. I say talk to the mother and find out WHY they have been late to school on such a regular basis, and get to the bottom of it. If it is because of unreliable transportation, then I would suggest having the child ride the school bus to school each day to make sure she's on time.
Find out why, and figure out a solution with you, the childs teacher and the mother, and the child as well.
I think after a conference the mother would get the idea, that no matter how trusting folks may be, that in todays world that wasnt the smartest idea.
Dixie