Based on what I read there may have been a misunderstanding on what the deposit was for. If you did not stipulate in writing that the deposit was non-refundable even if you did state it would be applied to the last two weeks of care then you may have to return it. What I am getting from what you wrote makes it sounds like the lady dragged it out in making a decision and finally you asked for some money to be paid for you to be willing to continue to hold the spaces open but what she paid would be used towards future care. She may truly have not gotten the idea that it was money she would not get back if she didn't start care.

You could try splitting it with her with the idea that you are now facing a loss of income from and give whatever date it was the deposit was made. Problem is the family made it clear they hadn't made up their minds yet - likely were waiting for the sister to decide if she would do it or not so you can only claim a definite booking of the spaces as of the date the deposit to hold the space to the start date was made. If you explain that you have now missed out on many people looking for summer care because she had indicated she wanted the spaces and are out the income from summer and explain that that is what the deposit covers and why caregivers ask for a deposit and hopefully she will understand and be sympathetic and agree to let you keep half.

Unfortunately this is the risk we take when we interview a family and then stop interviewing hoping they will take the space. That is over 2 months you could have been interviewing other families for summer care. You are also seeing one of the reasons caregivers do not like taking a family with more than two children because when they pull out there goes 50% of the income.