If breakfast isn't included in the price of your hotel, you have numerous options in Madrid.
If you want something sweet, we recommend trying the Madrid classic "porras" (like churros but longer and thicker). A good place in the centre is El Maestro Churrero (Plaza Jacinto Benavente 2). Coffee with churros or "porras" will cost you 3€, and chocolate with churros 3.60€.
If you're in the mood for something salty, definitely try Paraíso del Jamón. There are several in Madrid, though the closest one to the centre is on Calle Arenal 25. Order a ham sandwich (2.50€) and cafe con leche (1.25€).