One place that is good for families is called Casa Mingo, and it's on Paseo de la Florida, 34- walkable from the metro Principe Pio.  It's a very old restaurant that's quite well-known and is famous for it's cider-marinated roast chicken. When you walk in, you're seated at a wooden table with stools, among walls of bottled Asturian cider, and giant barrels. The menu has a total of about 10 items, including chorizo, salchichas (sausages), ham and other meats. Still, the roast chicken is by far the shining star of this place. It's AMAZING and cheap! Order a few bottles of cider along with a salad or two, and you're set. It's a great place to go with kids because the waiters and customers are laid back and don't mind the presence of children, the menu is simple and and it's not an expensive dinner! 

Pizza Marzano on calle serrano 45 is also a fantastic place for family eating. The menu comes in English, the food is really tasty and the price of the food is very reasonable. The restaurant has a buzzing atmosphere and the staff are very friendly. Make the effort to go there, it is well worth it, especially Sunday lunchtime.