Wow, yeah, that's pretty ambitious! You're gonna get on like this guy:
I would most definitely just focus on a smaller number of cities for a longer amount of time.. 3-5 days per city is usually good, sometimes a week for bigger ones, maybe less for the smaller... For instance, a proper round of London or Paris would take at least 4 days each, IMO, just to get a feel for them. Otherwise, you're going to spend more time on the train than in the cities!
Example: You can fold Versailles into the Paris part and make that 4 days - it's only a 45-minute RER train ride away from the city and you'll be back in town by the late afternoon.. No need to make that a separate day on the trip, as you'd have to go back to Paris to head off to your next destination anyhow.
Also, if you "base" in a handful of the bigger cities for, say, 5 days at a time, you can get day-trips to other nearby places on the cheap through your hostel, but have the luxury of not having to lug your backpack around everywhere. It makes for a much nicer, more relaxing trip, and you still get to see a LOT - and better yet, immerse yourself a bit in each area.