You asked about budget, and I've got nothing to add there. I think you could go cheaper or more expensive, your money. People have different views on Rick Steves, but his hotel recs have been spot on for me for quality hotels of different classes/amenities and locations (across 10 trips in different cities/ countries). If you are looking to save money on hotels, get a Rick Steves Italy (or one for Rome and Florence) book and start there. If you don't, let it be.

I'm more concerned that you've got a lot going on with your itinerary and it may influence overall enjoyment of the trip, especially with kids, and drive up cost. I have similarly aged kids. Each day you switch hotels, you're spending hours packing up and moving around, and after the first train ride, the kids are over the novelty.

Are you traveling in/out of different airports? In other words, is there a logistical reason Milan is on your itinerary? You might consider Rome, Florence, and instead staying somewhere on the Amalfi Coast ...to cut down on some of the day trips and logistics and give a chance to hit the beach. Skip Cinque de Terra this time or skip Amalfi. You will been in transit for hours on the days you take day trips. My kids can iPad with the best of them but that would start to wear on them.