Oh, boo. That is a junky situation.
CAN you take them to court? Yes, you can sure try. Is it going to be worthwhile for you? Maybe not. My advice would be to talk to a small-claims type lawyer. Unfortunately, the challenge with our career is that once our services have been provided, there is very little recourse available when someone stiffs you. For this reason alone, I collect my fees upfront. Without a signed contract, it just makes it that much more difficult.
It is too bad that this family screwed you in they eye, but truthfully, it happens sometimes. Hopefully you can fill that 2 day spot, or that it works out for you otherwise. Clearly they had no intention of being honest with you from the get-go.
I am not an expert though, maybe someone else has better advice? Luckily I have not dealt with this scenario personally, but only heard from other's experiences.