Where to spend a week with 2 kids (age 10 and 11)?

Depends on their interests, partially. And your tolerance for heat. Much of Southern Europe is super hot in July/August, which is ok by me but seems to be a real problem for others.

If heat is OK I’ll suggest Greece. Kids that age might find ruins and myths interesting (mine do!) and you can also go to the beach or other outdoor activities. A few days on an island or smaller town and a few in Athens. We just did Athens, Nafplio, and Hydra which was a good balance. Nafplio is charming and my kids loved climbing the fortress. Also an easy drive to Mycene. There are beaches in town, too.

You could also try to personalize the experience for them by making it a "roots quest" of sorts. If your heritage traces to Ireland for example you could spend some time there trying to trace where your ancestors might have lived, go to the town of Cobh on the south coast to see where your relatives probably emigrated from, etc.
Preparation for your trip could actually begin before you ever leave home by collecting information from grand parents and the family archives in order to accumulate as much background as possible - would make the time in whichever country you choose even more productive and interesting for them, and would have the extra benefit of getting them actively involved sleuthing around their own ethnic backgrounds.
And if they really get into it you could splurge on the DNA test at Ancestry.com.

Since you’ll be in Italy, Venice or Rome come to mind. Ostia Antica as a day trip from Rome is wonderful.

Look into medieval festivals around Tuscany and Umbria for the time period when you are there. For example, the Volterra Medieval Festival is usually the last two weekends in August, and it is a blast, with lots and lots of kid-oriented things to see and do. There are festivals in many smaller and mid-sized towns during the summer, which might be a wonderful adventure for you and your kids to enjoy, together.

Okay, so i am very biased so naturally I will say Budapest. I've also entertained kids in Budapest and I know it to be very kid friendly.

Ryan Air from Pisa or Bologna is under $75 with a suitcase and a reserved seat. Nonstop in 1:30 to 1:45.

The lower cost of tourism in Budapest will pay for most of the ticket cost; and you get a totally different cultural experience from Italy.

For the kids
http://visitbudapest.travel/guide/budapest-with-kids/
http://travelwithbender.com/travel-blog/hungary/10-things-to-do-with-kids-in-budapest-hungary
https://trip101.com/article/family-fun-20-things-to-do-in-budapest-with-kids

I didnt see the circus on the lists (?) Great time at the circus. https://www.fnc.hu/eng/

The Dolomotes should be pleasant, temperature-wise, if you stay up in the mountains (Ortisei is often recommended here). There are lots of easy as well as more challenging walks, plus cool mountain lifts. You could day-trip down to Bolzano, which has the museum holding the Iceman as well as a drop-dead-gorgeous historic area. Bolzano is not where you want to spend the bulk of your time because it's down in the valley and can get very hot. Also, a lot of the moderately priced hotels do not think they need air conditioning. They are wrong.

I think the children would like Venice, but I'm worried about the weather. You may already have had a week of high temps in Tuscany

Consider getting the kids very involved in the planning related to the extra week. If they have input into the locations and activities they'll be much better and happier travelers. That in itself will make things a lot easier on you and a greater trip for all.

Scotland of course

I tend to look in the same area where I've already arrived. There are festivals in the Tuscany area in the summer that might be really enjoyable (I think July is the biggest month). Also the lakes up north. If not that, consider something to cool off as Italy will likely be warm to hot by late June. The Dolomites is a good choice if you like mountains. If not, the coasts are another good idea. CT isn't far but is just okay, there are limited hiking and swimming opportunities (IMO not enough to fill a week). Other coastal areas include the Italian Riviera to the west or the Adriatic coast to the east. Ancona has ferries to different places that might be enjoyable like Hvar and Zadar.

Consider Switzerland, especially the Lauterbrumnen Valley in the Berner Oberland area. There are many villages, scenic hikes, lifts, waterfalls, and train rides. It is beautiful and could be cooler than other areas in Europe.

Well having been a child taken to Europe ( started when I was 10) and having taken my own kids to Europe ( 11,13) , kids like beaches , castles and ice cream .

I would skip big cities , my kids like seeing the coliseum in Rome, but for like ten minutes , my daughter liked Venice , but even now at 22 yrs old , assures me the two nights we had there when she was 11 were more than enough .

Switzerland is actually a good option , while not about beaches , it is stunning visuallly and even kids react to it that way , it is however a pricy destination .

Greece is wonderful , I simply can’t see a downside to a week on the islands ( I’d skip Athens with kids ) , you can get flights from time easily to several of the more popular islands . The people love kids , the hotels and food are inexpensive, the beaches are lovely, and you can build in some sightseeing ( ruins etc ) but still have time for just relaxing on a beach .

Actually you really can choose anywhere , just be sure to book your flight from north America as an “ open jaw” flight so you don’t have to double back , or fly into time or Florence etc then fly home from airport closest to where you end up ( ie fly home from Athens or Zurich etc ) .

Inter europeon flights are cheap and easy to hop , we’ve used Ryanair , Easyjet , Vueling , and several other ones . Just learn to read websites on luggage and check in conditions .

We’ve spent less than 140 bucks to fly from London to mykonos Greece last year on Easyjet !! Only 4 hours , and so many flights are going to be much shorter !

With two children—Sicily: beaches, Greek temples and amphitheaters, Roman villa, Mt. Etna, hotel with a pool.

I would look into the Salzburg area. Completely different culture, landscape and food, and very kid friendly. We spent 5 nights there with my 11 and 14 yo boys. Just a few ideas: mountain coaster near Hallein, salt mine tour, castle and ice cave near Werfen, Konigsee and Berchtesgaden. My kids liked the Red Bull hangar in Salzburg. This area is spectacularly beautiful with lots of outdoor activities, great public pools and water parks.

We have discovered that our kids appreciate a down day in which they are the focus. We try to find parks and playgrounds everyday with Gelato thrown in for good measure. Kids love the mountains and hikes. They are now required. I have found getting them involved helps. Start learning the language. Once we decide to go, we start learning the language. My oldest has to order a local meal from a menu in the local language and has to eat whatever it is. We try to find kids books set in the cities we visit and have her read them. It increases her interest. Visit your most import sites in the morning with them. They tend to tire out during the day. No electronics unless for pictures and after returning to apartment or hotel.