OP -- If it's in your contract, then please don't let it go. Don't listen to the people saying you don't deserve to be paid because you didn't work. If it's in your contract, end of story.
I recommend calling the following organizations for help:
Georgetown Law's DC Law Students in Court: http://www.law.georgetown.edu/academics/academic-programs/clinical-programs/our-clinics/DC-LSIC/staff-contact-us.cfm
This is a great place to start. Contact them.
Legal Aid: http://www.legalaiddc.org/
I don't think they handle cases like yours, but it's likely they will be able to refer you to people/organizations who can help.
Remember to save
all past and future communication with your former employers.
Hopefully this won't have to go to small claims court because that can be a stressful process and take time. I would suggest that, moving forward, all communication is in writing, and that you make sure you are presenting yourself as level-headed, firm, and confident. You need to show that you're serious, that you're doing your research, and that you're willing to proceed forward. My guess is that once the family realizes they won't be able to easily cheat you out of the money, they'll pay in order to avoid the hassle.
This can be tough. They might get mean. But hold your ground. The contract says what it says. If they thought it was unfair (it's not), they shouldn't have signed it. Unfortunately, some people are just assholes. Don't back down.
Good luck!