The fact that they are coming from the nanny route is another reason to give them the benefit of the doubt in the sense that it is all new to them and they will have to master the differences - ie who is the employer, who make the rules, etc.
One option would be to contact the family rather than wait for them to contact you. Do it with the idea that you feel things got off to a bad start due to misunderstandings and that nanny care and daycare are different and that you think it might be a good idea to start over. To have a meeting to go over the handbook again so that you can explain how it works and explain why.
That has you coming off taking the high road - note you still haven't committed to letting the fee slide at this point just a willingness to discuss it amongst discussing other things. It will be up to the family to decide if they want to procede.
Just as we take new parents through everything with a lot of hand holding it will be double for this family because you are not only going to have to teach them new habits you are going to have to unteach the old habits. But if you think the family has potential then it is worth the effort and especially if times are tight. Doesn't mean you need to let them walk all over you but is the future income worth the effort of dealing with what just happened.