That is a tall order for Milan---hotels there are expensive, especially close to the Duomo. I think you may have to compromise on the location to get a family room in your price range. for example, Una Hotel Century (a modern business-style hotel) will accommodate a family with 2 children in a junior suite for around 150 euro----but it is close to the train station (Milano Centrale), not to the Duomo. Hotel Cristoforo Colombo (a Best Western) is in a better area, but still away from the Duomo. They also have family rooms, but you'll have to e-mail to get the price.
With children that age you might like to be close to the pedestrian street, Via Dante. You could try Hotel London and Hotel Star to see if they have family rooms.