A dry cleaner's is called "una tintorería" or "tinte" for short. Dry cleaning is called "limpieza en seco." They're all over Madrid, although I'm not familiar enough with the Huertas area to give specifics there. I can't imagine you'd have to go far, though.
As for prices, they're not expensive. Here's a website for a tintorería in Madrid, and they list prices under "servicios." (They're still in pesetas instead of euros, though.)
http://www.comendadoras.com/tina/ Some tintorerías will do home pick-ups and deliveries, which is real convenient, but a bit more expensive.