Here are two recommendations, not too far from the above area (as discussed, easily reachable by Uber or Spanish Uber "Cabify" - both should be 15EUR or so from your hotel). If you take the airport bus to Avenida de America (if you are a family, better and probably cheaper to get uber/cabify instead) beware of pickpockets in the airport bus.
- Restaurante Canadio. Conde de Penalver 86, Madrid. This place is quite good, more seating down style as opposed to informal/standing tapas bars. Perhaps a tad pricier than the average Spanish place but you get great quality.
- El Rincon de Jaen. Doctor Gomez Ulla 6. This one is more informal (and cheaper) but has both standing style kegs with stools as well a few tables in the back where you can sit down.
NB: Not sure how well they will be able to speak English to you, particularly in the second place. There may or may not be English menus available so if you have some knowledge of Spanish, that would help.