Best/most affordable transportation from Amsterdam to Florence or Rome

Easyjet flies once a day from Amsterdam to Rome for 38 to 75 EUR pp, depending on the day.

Definitely fly to your third destination to save time. Also, assuming you have 9 nights actually sleeping in Europe, I guess I'd divide it 3-3-3, but it's going to feel a little rushed, but doable. You could also just do two locations and get a bit more in depth.

If you haven't been before, Paris needs 5 full days minimum. 3 countries in your timescale is too rushed as you will lose at least half a day in transit.

Flights for this summer (you don't say when you are travelling) have been available for months and prices only ever head in one direction. Fly open jaw to maximise your time.

Stick to Paris and Amsterdam and cover both properly. Rome needs 5 days, so this would be too rushed.

Keep in mind that the number of days you spend anywhere is personal opinion. I always think that at least two nights is a good starting time for major cities like London, Paris, Rome, etc, keeping in mind that you could easily spend a week, depending on what you want to see and do, in most cities. Over the years we have spend nearly three weeks in Rome, a month in London, a couple of weeks in Paris but will be returning next Fall to both Paris and London for a few more days. Always something new to see and do.

An travel to Italy will involve a much longer travel day. I would like at a tighter triangle - say Paris, Bruges, Amsterdam. Fly into Paris and home from Amsterdam or reverse. Same Italy for another trip.

My rule of thumb for how long to plan for any stop is to plan two big sights a day, one in the AM one in the PM, with lunch between. You may have time to see more but it's a mistake to plan for it. One full day in a destination is flanked by two nights. Two days needs three nights. Don't plan one day equals one night. You'll lose at least half a day every time you move to a new destination; more when you make a big jump (like Rome).

Also, if your daughter is a minor, make sure you have plenty of documentation that she is yours and you have permission from the other parent to take her across a border crossing

How old is your daughter? If she is 18 the problem of approval from the other parent or guardian does not apply. If she is under 18 it only applies for the flight to and from Europe. The other trips are within the Schengen Area and there are no passport checks.
As she is young (I assume), 3 x 3 nights will not be (for her) too rushed. Think of it as a teaser so she wants to come back again. For me, I would skip Amsterdam and make it 5 + 4 nights.
Your choice of Italian city will be influenced by where you can get a flight back to from wherever you live.
I would suggest Rome or Venice. A flight from Amsterdam to either would be easy. Both Venice and Rome are "mind-blowing" in the sense that until you have been there you cannot imagine that such places could exist. If you live there, they are mundane. If you do not, it is difficult to describe the concept of a 500 year-old (or 2000 year-old) building at the end of the street, that nobody has bothered to re-develop in that time, and is not exceptional for the city. You do not realise you have your concepts are so restricted until you have seen what lies outside.

Fly to Rome from Paris or Amsterdam. If you can return home from Rome that would be the most efficient without backtracking. 3 nights in Each location minimum. Amsterdam is easiest to start a trip.