I had a great 3-night stay at Fernando and Patricia's place.
It's situated about 10 mins from Atocha station which is perfect for the travel to and from the airport, and only about 15-20 mins walk into the main city.
The room is a great size, is really quiet and shutters ensure a fantastic night's sleep. Having a private bathroom is also a nice addition.
Both Fernando and Patricia are extremely friendly and I really appreciated their recommendations.