You can fly to Bologna (via some European hub, like Paris or Frankfurt or London etc.) then take the trains down to Florence and Rome. Florence is between Bologna and Rome approx. 35 min south of Bologna via high speed rail. Then you can fly back home from Rome.
Using Kayak or Google Flights enter a multicity flight
Going:
From YOUR HOME AIRPORT
To BOLOGNA (BLQ)
Returning:
From ROME FIUMICINO (FCO)
To YOUR HOME AIRPORT
I don't know how much the flight cost would be, it depends from where you fly from etc.
The train from Bologna to Florence is about $35 full fare in standard class.
The train from Florence to Rome is about $75 full fare in standard class.
If you purchase tickets in advance (if you know the exact dates and times, since discounted tickets are not refundable/changeable) you can save up to 50% and more.
Cities like Florence, Rome, Bologna can be expensive, but you can definitely find a hotel room for under $200/night in a nice 3-star (or maybe even 4-star) hotel even in those expensive cities. That generally includes breakfast, which is often buffet, therefore you can easily skip lunch if you load up at the hotel in the morning. For dinner at a regular sit down restaurant, you can budget about $35-40 per person, inclusive of multi course dinner, dessert and wine. Of course during the day you will have gelato, snacks, drinks, bus tickets and I imagine museum tickets, but it shouldn't be more than $30 a day for all of the above. So you can comfortably survive with $250-$300 day (or less if you downgrade your hotel accommodations). So $5000 will be more than abundant for 2 weeks (exclusive of airfare).
The challenge is to get to those smaller towns. Sulmona and Castel di Sangro, are both in Abruzzo and relatively close to Rome via train. The train would be inexpensive, since it's regional commuter trains. The two towns are not far from each other and connected by bus, although not frequent. The hotels would certainly be under $150/night. The most efficient way to reach those localities would be a rental car, but that would increase your cost since rental cars in Italy are $40-50 a day. Gasoline is expensive (about $7 a gallon), but the distance wouldn't be too big and freeway tolls would be minor also.
I think you might need to skip Sicily for this trip. That requires an extra flight from Rome and you definitely need a rental car to go to Villarosa. Since you are 70 years young, you have time to go again another year so that you can concentrate in Sicily alone. The island is so rich in places to visit, that it will keep you busy for another two weeks.