It sounds as though you have been fair and you should listen to that little voice inside.
I would give them either one or two weeks notice so they have the time to find someone else. I always try to remember that we never know what goes on in someone else's life and what situation they are in. Having said that, you don't need to help them either but that would just be me.

Be firm in telling them that they can not arrive until the set time, payment has to be paid in full now for the reamaining time in care and if they are unable to do it then you need to terminate right there. You are giving them notice, helping them out as you can continue on without their income and since they have already started about money, don't wait until the last day as again, you don't know their situation and might take off and you won't see a cent.

Good luck- let us know what you plan to do, not easy