RS offers "family tours" - 13 days from London, through Paris, the Alps to Florence. Family tours have mostly kids and their parents, fewer retirees. The downside for you is that they are offered only from May to July. Tours manage to see so much in that time because everything is well-organized, dawdling and being late are not acceptable (and not imposed by parents!), transportation is door-to-door for hotels and some sights, when that's not possible the guide knows the way so there are no wrong turns or wrong trams. There's usually no waiting in line because everything is prearranged, including meals at restaurants. Tours don't usually spend more than a couple of hours in a museum and it's always possible to opt out of an activity, your guide will help you plan an alternative.
Here are some things to consider. How much time is in 2 weeks? Let's say you take two calendar weeks plus an additional weekend. Fly to London on Friday evening (which I won't count). You land in London in the morning, but since the flight is only about 7.5 hours. After drink and meal service, you'll have maybe 3, at most 4 quiet hours before breakfast service. So you are not only jetlagged but very short on sleep. So Day 1 doesn't really count - you're zonked, maybe someone's cranky. If you're lucky, everyone is excited to start a new adventure. Get to your hotel, check in, maybe unpack and have a shower. Then head out to explore. There will be culture shock and disorientation. Signs are different, money is different, enough words are different to cause a little more confusion. never mind traffic rules and tiny, twisty streets. Try to stay awake until evening so you have a better chance of getting a night's sleep. Day 2 is often a foggy day as well, as you get over jetlag and catch up on sleep. Being on your own in London means learning how to use public transportation. The Underground is easier than the el, but coming out of the station can be disorienting. Buses do not stop at every block and the routes make lots of turns.So that's 2 days in London and you've seen walked by some of the sights. Stonehenge is a day trip and you'll need to take a tour.
Count nights, not days. You need one more night than day to have a full day in the city. (The rule is X+1 Nights = X days). Changing locations will use up at least half a day. It's about 4 hours by train from London to Amsterdam, then add time to pack get to the train station, then get to the hotel in A'dam, that's easily 6 hours. Amsterdam means a new currency, new transportation, in other words, getting oriented all over again, this time with a new language thrown in.
Florence is about 850 miles from A'dam - going by train will take well over 12 hours, maybe as much as 20+, with 2 train changes. If you keep Italy on the agenda, you'll have to fly.
So Day 1 was Saturday arrival in London. Let's say you fly home 2 weeks later on Saturday, Day 15. You'll probably arrive back in Chicago Saturday evening, giving you Sunday to adjust somewhat to the time difference and the jetlag. That gives you 13 full days on the ground (Sun-Fri). With your proposed itinerary, you'll have lost most of 2 days in transit, another 1/2 day getting to Rome. That gives you 10 full days, split among 4 cities.
January is not a good time to go. The weather is likely to be dismal and in the UK and Amsterdam days will be shorter than Chicago. London is about the same latitude as Calgary, A'dam is even farther north. October is a much better choice for those two countries. But don't expect them not to be crowded, both London and A'dam suffer from high tourism even in October. Of the months you suggest, I would choose March for Italy. The weather is generally mild and often sunny.
All in all, I suggest you either most of your time in the UK with a few days in Amsterdam at the end, spend it all in Italy or maybe start in London, then fly to Italy.
