If you haven't already booked Carlton. This hotel seems to be in the area you want:http://es.melia.com/hoteles/espana/madrid/tryp-madrid-atocha-hotel/home.htm
But it only just scrapes into your budget in October.
If you take a look at google maps you can find a large selection of hotels based on area:
I hope that helps.
It'd be easier to find something in your budget if you let us know when you're travelling.