Unfortunately you'll need to take bus, train or plane to DC from JFK. There are flights to the two DC area airports from JFK on most major airlines. For train you'll need to get to Penn Station for Amtrak to DC (Airtrain to Jamaica Station and E subway or LIRR to Penn Station) or there are a few different bus lines to DC. Most leave from Port Authority Bus Terminal (E subway to 42nd St.).
Unfortunately you'll need to