My first thought is too many places, not enough time. It will all be a blur.
Typically, 2 nights = 1 day in a place. With 13 nights you should have 12 days of time on the ground counting arrival day but not departure day.
Your arrival day will not be all that useful due to jet lag, so I'd subtract it. That leaves 11. Going to 5 places requires 4 travel days, even if the distance is short. So that leaves a total of 7 truly useful days.
All of you will probably take about 3-4 hours to get up, get dressed, get packed, have breakfast, check out, get to the train station or airport and wait to board. After arrival, you'll probably take 2 hours or more getting to your lodgings, checking in, getting settled and getting something to eat, even if it's only a snack, before you can do anything else. None of that included the actual time in the air or on the rails. And keep in mind that the group will travel at the pace of the slowest member.
However, if you only go to 2 or 3 places, it cuts the travel day losses. It also makes renting apartments more attractive, which would probably be better for the whole family. Check booking.com for options, but family rooms or quads aren't that common.
It sounds like London and Rome are givens to meet your wife's wishes and part of yours. The other logical stop is Paris. I'd skip Venice and Florence this trip. If you choose London, Paris and Rome, I'd make it 5 nights in London and 4 each in Paris and Rome. And Eurostar between London and Paris, then fly from Paris to Rome.
If you choose London and one other city, I'd split the time between each however you want, and perhaps take a couple of day trips out of each. Obviously, if #2 is Paris you'd Eurostar and fly home from there. If #2 is Rome, flying from London would probably be the most efficient.
Once you and you wife decide which cities, then I'd get the kids involved in the planning for what to see and do, so that they are as invested in the trip as y'all are. Our son took his daughters, ages 10 and 12 at the time, to Paris. The 10-year-old avidly participated in the planning. The 12-year-old was like whatever. Guess which one had the best experience?
The Man in Seat 61 is THE source for all things train. There are lots of lodging resources, but I use Booking.com to identify potential places, then contact them directly. When you put in your family details, your results will predominantly be apartments, but you can refine your results in many ways, so it will take a little practice to learn how to use it.
Finally, everything will go much more smoothly if you all pack light using carry-on only. If each person, including the kids, can't manage their own luggage, they're taking too much stuff. People in Europe have to do laundry or have it done for them, too, so you don't need to pack more than a week's worth of clothes. Most apartments will have at least a washer and a drying rack. That's a huge convenience, especially with kids.
Have fun planning and have a great trip.