I am a Mexican Service Designer, I rent my own flat in Madrid when away for holidays I've lived in London for a while and now live with my boyfriend here.
Our flat is small but comfy and handy with everything interesting within 5 min walk distance, the Prado, Thyssen and Reina Sofia Museums are all within 2 blocks around. There is a very good Market Anton Martín and the Art Decó Cinema Doré is a must, if you want some nature there is an amazing wall garden in this new cultural space called the Caixa Forum that also has great exhibitions just right in front of the Botanical Garden and Prado. If you are not that much into art, there is also Medialab Prado, which is an open lab for people who wants to develop technology related projects, everything is for free and open.
There are all kind of cafés and bars in my area, from jazz to flamenco, amazing galician restaurants with loads of seafood, vermut from the tab is very common, I can help you out with all necessary info, map, free hours for the museums, cheap breakfast places, list of favorite local, hope you have a nice time in this city.
Oh! I almost forgot. Atocha train station is just right 3 blocks away, it has a train service to the airport that is 2.50 and a direct bus service as well that is 5€, the bus is between 30 and 45 min ride.