I did a high school exchange program myself. This was in the mid 2000s so I'm sure prices have gone up, but it was about 6K + airfare, insurance and pocket money. I paid for the program, but my mom bought airfare, and I had (very limited) pocket money by tutoring two classmates for 5/euro an hour once a week. It was a great experience, I'm sure it helped me in college admissions, and I just took my one-year-old to go stay with his "host grandparents" for a week this past summer.
I think your daughter needs to have skin in the game both economically and otherwise. If your daughter can start saving even a grand a year starting next year, she'd be able to make a dent in the tuition herself. Paper routes, babysitting, tutoring, whatever. Have her start researching school schedules in Italy because they're different than here. Have here figure out how studying abroad would affect her graduation here (would she miss courses she needs to graduate? if so, can she take a community college or online course to make it up? My state required a certain number of health/PE credits and wouldn't accept foreign equivalents). What about PSATs/SATs/ACTs? How will she take them if she's abroad, or plan so she doesn't have to?
Frankly, I don't think 10K is all that much money when you compare it to college which is a couple years after that on the horizon. What's your approach to college planning? Can you just roll part of this tuition in to your plan for that?