... simple Google search, huh? :))).... https://www.google.com/maps/search/lavanderia/@40.4172921,-3.7060251,17z
The closest are Pressto --an international chain of dry cleaners in several countries-- and, to the south, Tintoreria Madrid-Sevilla, a more traditional service, for nostalgics of a more personalised service. Also, I believe they also offer an ironing service -Pressto certainly does.
Note that none of these two are coin-based though, for those you'd have to walk further (ie: Lavandería automática Cervantes) or perhaps to laundry services at other areas (ie. La Latina). The coin-based services tend to be available in less "affluent" areas, so to speak.
As Ann said, if (if!) the hotel offers this service -not many hotels do- it's likely to be costly.