I would consider two, two-week stays. That would make local day trips
more convenient.
I'm in Frank's Camp here, Francis. With the sort of time you have, you could easily make one location change to put day trips within a different region within reasonable reach.
A couple of questions? Could you share the ages of your tribe, and what sorts of 'amenities or things to do' you're looking for? Your family's collective interests can make a difference where to hang their hats for 2.5 weeks.
That said, I'll vote for Florence, Italy and maybe a smaller town in Northern Italy. Padua? Ferrara? Will have some other posters weigh in on the North. Florence has a ton of interesting attractions IF they're of interest to your family, and a good number of day trips within reach; Pisa, Siena, Lucca, Arezzo, Pistoia and some others. While I wouldn't go there on a weekend, you could do a night or two in the Cinque Terre for some hiking and swimming, if the season allows, or even go to Rome. Florence itself is heavy on Renaissance art, architecture and history - things which appeal to me - but may not to you?
Affordability: that's a tough one as I don't know what that means to you? Housing for 9 won't be inexpensive. Train tickets for 9 won't be either unless it is just short trips by regionale rail; those fares are reasonably inexpensive. 'Things to do' often have a price tag: museums and whatnot almost always have an entry fee, Italian State Museums are generally free for visitors under the age of 18, and civic museums may also be free or offer reduced ticket prices for minors so that helps. As well, not all but many glorious Italian churches are free for the looking and may contain priceless artworks; this is especially true of Rome....
...which is another possibility. It is a fascinating city with a great many corners - free and not - to explore. As previous mentioned, there are tons of those free glorious churches, many of them very old. Spend a day trekking an expanse of the Appia Antica and the many ancient, crumbling monuments along the way; free to do unless you want to get inside, say, one of the interesting catacombs along the way. Go to Salerno for a couple of nights to see Pompeii and Herculaneum and get a look at the Amalfi Coast. Go to the hill town Orvieto for a day or longer...
What I personally wouldn't do is book a villa or agriturismo in a rural area. While nice for a large family, and they offer things like pools and cooking classes on site, they are usually not good choices for taking lots of day trips as they're often not on train lines, and may only have infrequent bus service. Rental vehicles are really a must for most of these.
Anyway, another couple of possibilities to add to the pile. :O)