Hello John L.,
In Oct. 2002 I stayed at
Hospedaje Madrid . It is located on Calle Esparteros? I believe. This is between Sol and Plaza Mayor. Can't get much more central than this. If you do a search, you can find my review which is better than I can remember about it right now.
I remember the price was around 90 Euros a night, it had a complete kitchen, a good size bedroom and washing machine. It was a little noisy, but being on the fourth floor, not really bad by Madrid standards. And most important, it had a good working elevator.