Hi, Terri. I think with 2 nights to spend and a flight that does not leave until 10 am, you could spend those 2 nights at a central hotel in London. I really like Premier Inns as they are very reasonably priced and would fit within your budget. I have stayed at a number of them throughout the UK and they are very clean, very comfortable and consistently good. You can find them all over London, but there are some that would be better for you given your circumstances.
Paddington Station fits that bill, I think, and there is a new Premier Inn that just opened up in that area. It is the Premier Inn Paddington Station, and would give you easy access to Heathrow, in that it is right next to Paddington Station. You can easily to to the sights from there. Yes, the Heathrow Express is more expensive than other modes of transport but it's very quick (as noted above by Camborne), and would definitely get you back to LHR quickly on the morning of your Italy flight. FYI, there is another Premier Inn nearby called the PI Paddington Basin, and is about a 5-minute walk from Paddington Station, so that would also be an option. But the newer one is closer and has gotten good reviews from those who have stayed there.
You could certainly stay at the airport, but I don't really see the need for it, as you have 2 nights in London, and it would take time and money to go back and forth.
ETA: Breakfasts are not included in the price at PIs but you can buy them separately and they are usually quite good. They offer a continental breakfast as well as a full cooked breakfast buffet for a slightly higher price. They also offer flexible prices for their rooms in that the highest price will be fully cancellable and the lowest will be non-refundable. Sometimes they offer a hybrid price where you pay upfront but the room is cancellable up to 28 days beforehand.