Hello, I recommend http://www.homesfortravellers.com

They have nice apartments at central Madrid at a very good rates and the staff is friendly and helpful. The apartments are fully equipped, even with Internet connection!!!

In the website you can see many photos and the map of location of each apartment and you can book them online. I hope you find useful this information.

Do you know of any that do apartments or houses in the mountains of Madrid?

Good information!
I also have useful information about apartments in Madrid. Apartments in Madrid have accommodations for every type of holiday, short or long stays. Enjoy your stay in our apartments and tourist hostels in our luxurious bedrooms. For more details you can visit us http://www.apartamentosmadrid.com