If you're on a budget, you won't find anywhere better to stay in Madrid than Hotel Regina. It's on Alcala Street and you can walk to attractions like the Puerta del Sol, Gran Via, etc. It;s also very close to the metro.
I stayed in a superior room and it was very modern with really nice decorations. The bathroom has a big walk in shower stall. There's also free wifi.
My only complaint would be that the walls are a bit thin - I could hear other guests in the neighbouring rooms.