We had a very enjoyable stay in the middle of many restaurants and sights at Hotel Roommate Alba. It is a very small chain. Here's Booking.com's description:
"Ideally located in the center of Madrid, Room Mate Alba offers air-conditioned rooms, a fitness centre, free Wi-Fi and a bar. This 4-star hotel offers a 24-hour front desk and concierge services. The property is non-smoking and is located less than 1 km from Plaza Mayor. Rooms include a desk, a flat-screen TV, a private bathroom, bed linen and towels. All include a wardrobe and a kettle. Room Mate Alba has a sun terrace. Popular points of interest near the apartment include San Miguel Market, Puerta del Sol and Reina Sofia Museum. The nearest airport is Adolfo Suárez Madrid-Barajas Airport, 13 km from the property."
For mid-May it's @ $220 per night for a double room.