Sistine Chapel

That is possible when you are not stopping to look at stuff. There may be a lot of human traffic (do you are you going to have a timed entry? mine was at 9:30 otherwise it's along line to buy a ticket the day of). A good plan maybe to store your luggage at the train station Termini.

Short version: please don't do this.

Long version:
1) You didn't say how old your kids are, but if they're "too young to appreciate" the Vatican Museums, they are probably too young to "appreciate" the Sistine Chapel. Do they have a special interest in it (say, because they studied it in school), or will it just be "a nice coat of paint" to them?
2) The crowds in the Vatican Museums are not to be believed - and neither is the tension. Imagine a mob of people, from all over the world, tense and frustrated because the Sistine Chapel is a long slog from the entrance, so they are all racing, pushing, and shoving to try to get to the Sistine Chapel as fast as possible.
3) Once in the Chapel itself, the guards are busy shushing everybody all the time. The crowd quiets down temporarily, then the dull roar of a thousand people whispering starts again, then the shushing starts again. It's anything but a "spiritual" experience, and unless your kids have a special interest, they will probably want to get out of there as fast as possible (I know I did).
4) Even if you make it to the Sistine Chapel by, say, 10 AM, you then have a long slog back to the museum exit (through huge crowds, again), and presumably have to pick up your bags at your hotel and then get to the station. So, much of your visit, even it were more pleasant, will be spent looking at your watch and worrying about time and logistics.

If your kids really have an interest in the Sistine Chapel, and if the only time to see it is the day you leave for Florence, then book one of the early admission tours. These are offered by the Vatican itself, as well as private companies like Dark Rome and Walks of Italy (The Pristine Sistine). These start at 7:30 AM, but you get to see the Chapel before the crowds, and before going in you get an explanation of what you are looking at (no audioguides, even whisper ones, allowed in the Chapel itself, so they give you the talk beforehand). This way, you and your family can actually enjoy the experience and get something out of it, instead of just "checking the Sistine Chapel box" and being tense and frustrated in the process.

Extremely helpful! Thank you! We'll probably just save it for a future visit.

Harold's reply is useful and 100% accurate, but I don't completely agree. You should not do this on departure day, and you should absolutely not do it without a prepaid, TIMED, admission ticket. This is something of a waste of money and trouble if all you see is the Chapel. (Would your children like the garden tour by tram bus?) If you can do this another day, without the time pressure of the train, it's not insane, just optimistic. And I'd combine The Vatican Museum with St. Peter's and perhaps Castel Sant' Angelo.

Vatican Museum tickets sell out well in advance. You should not go to the Vatican Museums without an advance ticket, anytime.

I have seen the incomparable Sistine Chapel twice, once before the Japanese cleaned up the art and once after.

The first time, my 7 1/2 year old Son was in awe of the Chapel. Previously, on our European trip, he was rather bored going through museums in Rome and later in Florence and Paris.

When we entered the Chapel you could see his face light up. He wanted me to sit on the floor and explain everything on the ceiling and the Last Judgment. I had a guide book and we spent from 45 minutes to an hour there. He was not bored and later he talked about the Sistine Chapel so much. Don't assume that your kids will not appreciate this magnificent work of art.

Jeff, I see from previous post that you have a 5 and 10 year-old? Your older child would probably do OK but your younger one may be miserable. With the crowds that mob the place, very short humans have a claustrophobic view of a lot of adult backsides, and those backsides all too close to their faces although IN the chapel you'll be looking UP, of course. Add the need to hustle and, well, I'll echo Harold's advice to skip it until he/she is older. Also, just IMHO that forking over the price of an an early-entry tour would largely be a waste if all you want to see is the chapel.

Side note: the crowds at the Uffizi may be uncomfortable for your youngest as well.

We were just there with our 9-year-old the other day. The crowds were pretty crazy. You walk down stairs and corridors, and then they kind of herd you into the chapel. They tell you to keep moving, but we found it was possible to move to the middle and stand for a few minutes to take it all in. My son was surprisingly fascinated by it as well. He had an earbud in and listened to his audio guide - looking for everything it called out. We really enjoyed the museum as well. The kids audio guide was great, covering many of the more interesting highlights. Maybe save the Vatican for your next visit when you have the time to enjoy it.

Thank you, again, everyone. I appreciate all of your perspectives. We are trying to make sure the kids don't get "museumed out," so we'll probably skip the Vatican Museum on this trip, for the reasons you cited and because timing would be an issue. And they probably wouldn't be interested enough to merit spending the extra money for early admission.

We'll spend our last morning in Rome doing something less stressful.

Your advice was very helpful. Thank you!