If you had just a few days to see a city in Europe with a 10-year old, which would be your top choices?

When my daughter was about that age, I took her to Amsterdam. One of the reasons was that Amsterdam minimized travel time to and from the US which doesn't really apply to your trip. However, the Netherlands has a huge number of things to see and do and it's very easy to get around on the trains and buses. There are museums for just about any conceivable topic. There are several organized day or half day trips, but you can easily do the same things using public transportation - for example, the Madurodam in the Hague is interesting for kids (miniatures of buildings around the Netherlands). It's also on the tram line that goes from the train station to Scheveningen beach. If you go there, the aquarium is small, but nice. You can easily get to Zaanse Schans to see the windmills or to the cheese market in Alkmaar (Friday mornings). There's the Anne Frank House, the Rembrandt House, the Rijksmuseum, the Van Gogh gallery, or Ons' Lieve Heer op Solder. Take a canal cruise or visit a castle - Muiderslot was interesting.