It really depends how much you are trying to pack into the other days. Figure that flying or train to Palermo or Catania will occupy most of a day there, then another back. The very minimum time you would need to allow is three nights, four better, and that would get you one city, so easily, Sicily would take 1/4 to 1/3 of your trip.
Could you do it? Yes, but it likely means just scratching the surface of some of the other places, which is fine with some, not so much others.
My itinerary might be something like: Head right to Venice, spend 3 nights there. Take the train to Naples (little under 6 hours), do 4 nights in Sorrento, see Pompeii, a bit of Naples, maybe a bus ride to the Amalfi coast.
Plane or train to Catania, then on to Syracuse/Ortigia, spend 4 nights, fly back to Rome for the balance of your trip. (Palermo would be your other option, with maybe a beach day and a night in Cefalu, depending on personal preference)
Most on here will tell you you don't have time, that 17 days is not enough for even Rome, but it is your trip, if this works, then go for it. Just go in with the realization that in someplace like the Naples area, you are not going to spend a day in Naples, one on Capri, hit both Pompeii and Herculaneum, and lots of time on the Amalfi coast. Same with Venice, same with Sicily, and even Rome. You will have time to hit 2-3 top sites, and that's about it...but that is fine.
If you go with it, come back, ask about each city, you will get lots of help, just ignore the naysayers.