Even without a kid, it seems like a lot of day trips for just twelve days. You want time to enjoy Amsterdam and Brussels (or wherever you stay). I agree with the advice to get a car to explore the countryside, but I'm not convinced that there's a lot to see and do in Luxembourg unless you want to visit one of their (few) castle hotels with good restaurants. Is it on your list just to check off the box? I would also suggest arranging things so that you can see places other than Amsterdam (such as Bruges) at night when the historical buildings etc.look pretty.

BTW, I liked Rotterdam and a kid might enjoy the water there (canals, boat ride, tour of the big port).