If I understand you correctly, the current DCB will be there until the end of June? The infant will not begin at all, despite some verbal agreement that he/she would? But no contract?
Personally, I do not give refunds. When parents pay a deposit, they have that applied to their final 2-weeks upon a proper 2-week termination notice. No 2-weeks, they lose that deposit. Also, free weeks/vacation are not usable for a termination notice (although it just sounds like they were using it in general next week). So, personally, I would keep the child while enforcing my policies. Unfortunately, part of our business is competitive, and parents will always do what is best for them in the end. All you can do is stick to your contract and fulfill your end of it. Also, if you held the spot without payment, it is very unfortunate, but I don't think there's much you can do without a contract.
So sorry that you're coping with this; it is one of the parts of this job that I hate as well, but all you can really do is use it as a learning experience for the future. Good luck!