Where To Go With A 7-Year-Old

With 10 days, I'd say take her to Venice and then Florence on the way to Rome. They're three very memorable cities.
Travel by train would be quick, easy and inexpensive on fast trains between these cities.
From Rome, you could fly cheap on Vueling or Ryanair to Barcelona, and it's a beautiful seaside city. Just make sure you travel light to avoid expensive baggage fees.
You can easily get back to the U.S. on direct flights from Barcelona.

There are a million great place in Europe so if people take the time you should get a lot of answers based on what people know about. Keeping a 7 year old occupied while enjoying it yourself is the challenge. For one option, there are direct flights Rome to Budapest and they aren’t terribly expensive. Here are some links of things mom, dad and daughter might enjoy:
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http://www.gyermekvasut.hu/english/chrw_home.php
http://www.zoobudapest.com/en
http://www.szechenyifurdo.hu/
http://en.wikipedia.org/wiki/Margaret_Island
http://visitbudapest.travel/activities/fun-things-to-do/budapest-chairlift/
http://www.tropicarium.hu/Tropicarium-Budapest-Hungary
http://visitbudapest.travel/activities/danube-cruises/
http://riverride.com/
http://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Budapest_streetcar_and_the_Danube.jpg/640px-Budapest_streetcar_and_the_Danube.jpg
http://www.cntraveler.com/readers-choice-awards/europe/best-cities-europe_slideshow_1-Budapest_9

What does your daughter think? and what is she interested in? My eight year old son was totally into trains and buses so we went to England for the Tube and the double deckers ( this was a few years back ) and built a lovely trip around that , castles and harry Potter along with steam trains and York.

At 8, I loved Heidi and the Sound of Music --- so at 27 I took my first European trip to Austria and Switzerland and have returned again and again. I would still vote for that _Salzburg is a charming small city with lots of fun child and adult things to do - the Salt Mine tour with the slide and I cant remember the name of the gardens but they have trick water fountains that are fun.

We live in a big city so I try and incorporate some small towns or smaller city with nice parks so the pace is slower. My son told me he mostly remembers playing Uno while sitting outside the B&B in York.
He also did love the 'ghost' tours.

Have a wonderful time - I miss that age!

I think Paris is a good destination for children. It has a science museum for children : "children's city" in the huge Cite' des Sciences et de L'Industrie at Parc de la Villette. There children can experiment and play with machines that show how scientific principles work. I think a person of age seven is not too young for that. In Paris I guess you could find entertainment for children, perhaps a puppet show. I think a girl, age seven, would like walking in the Cathedral Notre Dame and Saint Chapel at Isle de Citie. And the Tuleries (in front of the Louvre) to see the many flowers. And Musee Jacquemart - Andre' : it is a big opulent house made in the 1890s, it has furniture from the 1890s, and several good old paintings by famous European artists. On the internet at www.musee-jacquemart-andre.com That house (mansion) is located near Parc Monceau (the most beautiful park in Paris). I am sure that other people reading your question could think of more places in Paris and near Paris, that are liked by children.

I have a 7 year old boy, and I've showed him pictures and brief descriptions of places under consideration. Perhaps your daughter will have a preference, after you gather some good options here. What does she like? My son has shown an interest in castles, and when he was younger trains were the thing.

Ideas, off the top of my head:

Paris is always a hit. The Eiffel Tower would be great, the parks are fun for kids, and there's a ton to do

London has the lowest language barrier, obviously, so maybe she'll enjoy meeting other kids in the park, there are kid friendly museums, and I've heard of little girls really loving a formal (ish) tea

I'm off to Spain soon, and from what I've heard, children are very welcomed there, supposed to be a very kid friendly culture

Somewhere in the Alps if she likes mountains and outdoor activities

Venice

I'm taking my daughters to London & Paris this summer. (Ages 10 & 7) We are visiting friends in London, but even with 7 days planned there, we won't get near to doing everything. I know we probably won't get the half the things I think we will, bc the kids will detour us. But it's their vacation too & if they aren't happy, no one's happy. I'm super excited about Paris. We will probably not hit the major museums. I'm thinking that by that point in our trip certain things will have lost their novelty. I'm trying to be selective about the museums, churches, & castles, as they will start to run together after a bit. We will be doing the standard Paris "must sees," but the things I'm most excited about? Versailles & Giverney. In Versailles, the palace is obviously super cool. We will be spending the afternoon picnicking & bike riding. This way, they are getting in some much needed movement, but still seeing some really cool history. Both my kids have learned about the impressionists in school, (I'm sure yours has as well, 'Art Vistas',) and I'm very excited to show them where Monet lived and the garden that inspired him. The one museum we will visit for sure in Paris is the Musee de l'Orangerie, where the Water Lilies paintings are located. I't s a small museum & I think they'll really enjoy it. Ok, my 2 cents on that one.

On a side note, after much looking I finally found some kids travel books. (Talk about finding the Holy Grail.) These are by Lonely Planet called, "Not for Parents London Everything you wanted to know" They make them for London, Paris, Rome or New York. They're great for getting your kid excited about the trip. Lots of pictures & bits of trivia, not planning advice. I couldn't find them in the library or local book stores and had to order them online. I know some of the review complained they were all over the place, but I thought they were perfect for the 5-12 age group. Have fun!