I’m in the exact same scenario, and spent lots of time trying to figure out the right, legal way to do this. The long and short of it is that you do need to have a license from the city for this (although in my experience many people do this without the required license). It’s not super easy to figure out but basically you need to apply for a business tax ID, then get a business license here: http://dcra.dc.gov/DC/DCRA/For+Business/Apply+for+a+Business+License/Residential+Housing+Rental+License+Information
From your description, sounds like you need a One Family Rental License. Hope that helps!