Madrid hotels for around US$200 a night. Any suggestions?

There are hundreds in that range, so you should have no problem finding one in the area you want to stay in. Last October 22-23 we stayed at Hotel Europa at the Puerta del Sol. For two nights is was €414. We stayed at the same hotel September 25-26, 2023 and the two night price was €241, so the date will affect the price.

If a major convention happens to be going on in Madrid when you will be there, hotel prices will skyrocket; double or triple the regular rate.

I stayed at the Oriente Palace Apartments right next to the palace last November and loved it. My apartment was very spacious and beautifully decorted. I paid €139 a night. I liked the location because it was still central, but not as hectic as the area around Puerta del Sol.