I've never stayed in the area but if Madrid Man doesn't mind, I'd like to suggest looking into renting with YesMadrid. I've booked with A great guy named Fernando for the last 3 straight years and in two weeks I'll be coming again, making it 4 years running. He's got flats all over town but we usually stay in the city center. This year we're booked in Fuentes 3. Last year we were in Sol 2.
The flats have wi-fi, a local phone # and good TV.
At least send him an e-mail and see what you think. Maybe even mention Rudy and Nancy from California. I'm sure he'd take great care of you.
We usually just tell him when we're coming and what area we'd like and he gives us a few options to choose from.
I have no affiliation with them at all. I've just been a VERY satisfied customer and wanted to share the experience.
Here's the web site. Good luck. I hope you find what you need regardless of what you choosehttp://www.yesmadrid.com/apartyesmadrid.c.html