There are many many hotels and "hostals" in this area at that price.......
My last two trips to Madrid, I stayed at
Hostal Adriano , and quite honestly, it is the only place that I would stay now. I send all my friends there, and two of them are going at the end of this week.
A hotel doesn't necessarily have better rooms than a hostal. In fact I have stayed in several whose rooms aren't nearly as nice as the Adriano. What you get from a hotel is more services, like a restaurant.....
I am sure that MM will put in a link to Adriano's website....

(done!)
You won't go wrong there, but many many other choices exist also....
Just look at other members' comments....
Have a nice trip.....