This will be an issue for all interviewing in the month of January especially anywhere around the week of the 20th (inauguration day). Everything will be inflated during the week before, the week of and after the week of inauguration.
I would hope that medical schools would try to reschedule interviews during the days leading up to that date but this may not happen. My advice would be to try to reschedule the interview, however I would hope that adcoms would reschedule all interviews in the DMV area because everything (even student's apartments) are being rented out at over $400 per night and up.
Just for some perspective, my roommates and I are renting out our 4 bedroom house for $3000/night during inauguration weekend... it's going to be impossible for you to find anything between Jan 15th and 21st; maybe even before and after that with the amount of new employees transitioning into the city.
You should call the school and try to stay with a student host; or at least find out what they'd recommend. Good luck on your interview.