I think the best bet would be something related to the medieval towns & some museums on that note (London could be a good starting point as it is well-known tourist attraction with some excellent museums on this theme but I'd go more for less known marbles) Or maybe you'd like to go further in history - romans&greeks? so Italy & Greece - Rome and Athens are well known tourist attractions. That would be something interesting for kids and not available in the US.
You should start browsing for tops X towns related to this subject. Something like:
http://www.bigboytravel.com/europe/topmedievalcities/ This would probably give kids of most ages and sex something interesting to see.
Maybe you can time it somehow to visit some medieval festival where they have battle recreations (for instance in my country Romania there's this one:
https://www.google.ro/search?q=festival ... AT8QsAQIOQ). Maybe the kids would like to play dress-up and try some costumes/armors if available and possible...(not sure if this is possible or not).
As a hidden gem ... If you want some time on the beach as well as something to visit I'd recommend Greece. Corfu was the best experience for me although you would need to spend quite a few days there if you go. You have everything there: long sandy beaches with warm shallow waters for tens of meters into the see (towards the continental Greece and Paris's Albania) and on the other side small and rocky bays with cold water (toward the Mediterranean sea). There's an old english fort, port and the nice town of Kerkira to visit as well as princess Sisi's summer vacation house(now a museum) and some interesting "top of the mountain" views over the island etc... I'd recommend a 1 day tour of the island and a 1/2 day boat trip to the nearby smaller islands if you do want to pick this as a destination.
Probably Italy's seaside would also be a good pick - Pasta might be able to help you out with this (?) and recommend some, not so popular, sites if you're not looking for the most renowned tourist attractions.
Spain would also be a good pick. Everyone recommended me to skip Barcelona but instead go for Valencia. Everyone that went there came back with great memories wanting to revisit it. I've heard it has everything you'd want in a vacation location: the beach, music, history, culture and food.
Oh yes... Do try the local specialties wherever you go. At least for me this is one of the must do in every vacation!
I also liked Holland. Loved to move around in the water-buses and quick tours of the city centers of Rotterdam and Amsterdam. I liked the old windmill towns and plan to visit the country again during the tulip & flower festival... but more for my wife than myself

. However, this might be a bit more tiresome than fun for kids ... depends on their age I guess.