I'm going to give this a try. As you will see below, there is a pattern. One day of exploring and one day of long travel. Essentially, 40% of your trip will be travel days. On these travel days you need to take much more into account than just the length of the train ride. Your time will also including checking out of hotels, getting to train stations, orienting yourself and finding your new hotel.
For me, this itinerary is the exact opposite of the way I like to travel. I do not scratch the surface. I like to spend a week or more in one place and make interesting day trips. I like to unpack my suitcase, put away my clothes and feel like I am living somewhere for a while. I like to find a nice corner restaurant that I go to more than once. I like to be somewhere long enough in case there is a bad weather day.
If I were you, I would stay in Austria. I would split my time between Vienna and Salzburg. The Christmas Markets will be happening and it is a magical time here. Save the rest for another time. Plenty to keep you busy for 10 days. Who know, you might even remember the place.
So here is how your current itinerary would go:
Day 1 - Vienna. Can I assume that you are arriving internationally? If so, expect to be exhausted and don't plan to accomplish much that day. Maybe walk around the center of town. Spend night in Vienna.
Day 2 - Full day in Vienna. Spend night in Vienna.
Day 3 - Vienna/Innsbruck. Wake up, have breakfast and check out of your hotel. Head to train station and catch the 11:30 am train which arrives I Innsbruck at 4:50 pm. Check into your hotel, have dinner and call it a night.
Day 4 - Full day in Innsbruck. Explore the town or the mountains. Not really time for both. Spend night in Innsbruck.
Day 5 - Innsbruck/Luzern. Have breakfast and head to the train station. Catch the 10:54 am train which arrives at 3:49 pm in Luzern. Find hotel, check in, have dinner.
Day 6 - Full day Luzern. Spend night in Luzern.
Day 7 - Luzern/Florence. Take the 9:40 train which arrives at 3:55 pm. Find hotel, check in have dinner.
Day 8 - Full day in Florence.
Day 9 - Florence/Rome. Check out, go to train station. Take the 10:38 arrive at 12:10. Find hotel, check in. Explore in Afternoon.
Day 10 - Rome. See what you can in the morning an head out.
Here is how I would do it:
Day 1 - 5: Vienna
Day 6: Vienna to Salzburg, train ride if 2 hours 20 minutes.
Day 7 - 9: Salzburg and surrounding area
Day 10: Train back to Vienna, go to airport to fly home.