Basically you have three hotel location options:
1. Puerta del Sol (the center of everything, no need in transport)
2. Railway station Atocha (all three major museums and Retiro park are there), 20 min walking to Puerta del Sol, no need in transport.
3. Salamanca (more quiet, high end residential area), metro access
Taxi to and from airport is inexpensive.
I suggest first two. Examples of the first two:
Toc Hostel Madrid Plaza de Celenque, Madrid City Center, 28013 Madrid, Spain (Excellent references)
Hotel Mediodia 2-star hotel Plaza del Emperador Carlos V, 8, Madrid City Center, 28012 Madrid, Spain
Catalonia Puerta del Sol 4-star hotel, Atocha, 23, Madrid City Center
Hotel prices are are very good for your dates: check out Booking.com. Don’t be afraid of small 2-3 star hotels or hostels if they have high rating (about 8.0 and a few hundred reviews).
Hotel Mediodia 2-star hotel Plaza del Emperador Carlos V, 8, Madrid City Center, 28012 Madrid, Spain
Three top museums to visit: Reina Sofia, Prado, Thyssen Borgemiska (within 2km radius). All three major museums have free visit hours. If you have time and wish – also a small Sorolla Gallery.
Reina Sofia Museum, Madrid
Prado Museum, Madrid
Thyssen Borgemiska Museum, Madrid
Two food markets: San Miguel near Plaza Mayor/Puerta del Sol (high end), or Anton Martin (closer to Atocha) – low end, which does not mean bad..
Shopping (including fantastic food): El Corte Ingles on Puerta del Sol.
You can spend a day in Toledo, but I would stay in Madrid all week.
