Denmark would be great family fare. Its landscape is dotted with things constrasting new and old with cities, castles, and quaint little villages on its many islands. You can give history lessons on the Vikings to the kids. The kids will also enjoy visiting Hans Christian Anderson's birthplace. And the little mermaid awaits in Copenhagen.
The original Legoland is in Denmark and the prices are very reasonable with a very family and kid-friendly hotel on site. You can book online from their website:
http://www.lego.com/legoland/billund/
I suggest you pick up a travel book on Denmark to check for yourself. Driving is easy there although there are many other forms of transportation. Everyone can speak some English there. I would suggest at least one week there for the highlights, two if you want to do it more at a more leisurely pace to visit the entire country.
When we visited Denmark a year ago, it also felt that there were less Americans than at other popular places in Europe. Don't mean to offend the Americans (I'm one too), but who wants to be surrounded by Americans when visiting a foreign country?
Another tip for the summer: The Danes make a special drink (non-alcoholic) from a wild flower called Hydebaerblomst. The ready-to-drink bottles only come out in the summer, as I was told. But the condensed version in big bottles (size of a wine bottle) can be found for very inexpensive prices in the supermarkets. I have never seen this beverage anywhere else but Denmark.
Also, let's not forget the necessity of life. You will appreciate the cleanest bathrooms in Western Europe. At least, that's what we surmised after sampling Denmark along with Germany, Holland, Belgium, and France.