All of it in 4 weeks, not a difficult task with unlimited funds.
Day 0 Fly to New York City (probably JFK from Europe) Day 0 or Day 1 depends upon how early a flight you can catch to arrive at JFK before noon? A Friday flight from Europe should get you to New York easily in time for a Saturday train departure on the first train of your train trip, or a Monday departure if you wish to spend some extra nights in New York City.
Day 1 (Saturday-Monday-Thursday only) {See Day 3 for reasons why}
Catch the Crescent to New Orleans which departs New York at 2:15 pm Eastern Time
Day 2
Crescent arrives in New Orleans at 7:32 pm Central Time. You will have to rest overnight in a hotel. Maybe a great chance to visit a jazz bar?
Day 3
Catch the Sunset Limited to Los Angeles which departs New Orleans at 9:00 am Central Time on three days only (Monday-Wednesday-Saturday only)
Day 5
Sunset Limited arrives in Los Angeles at 5:35 am Pacific Time
Catch the Coast Starlight to Portland which departs Los Angeles at 10:10 am Pacific Time. You will have an overnight stay in LA later.
Day 6
Coast Starlight arrives in Portland at 3:32 pm Pacific Time
Catch the Empire Builder to Chicago which departs Portland at 4:45 pm Pacific Time
This in only an one hour transfer - it might be safer to overnight in Portland, are you a gambler or not - if you are subtract one day from the rest of the trip.
Day 7
Catch the Empire Builder to Chicago which departs Portland at 4:45 pm Pacific Time
Day 9
Empire Builder arrives in Chicago at 3:55 pm Central Time. You will have to overnight in a hotel. Enjoy the Windy City.
Day 10
Catch the California Zephyr to Emeryville which departs Chicago at 2:00 pm Central Time
Day 12
California Zephyr arrives in Emeryville at 4:10 pm Pacific Time. You will have to overnight in a hotel. Enjoy the Bay Area.
Day 13
Catch the Coast Starlight to Los Angeles which departs Emeryville at 8:20 am Pacific Time
Coast Starlight arrives in Los Angeles at 9:00 pm Pacific Time. You will have to overnight in a hotel. Enjoy LA.
Day 14
Catch the Southwest Chief to Chicago which departs Los Angeles 6:00 pm Pacific Time
Day 16
Southwest Chief arrives in Chicago at 2:50 pm Central Time
Catch the Lake Shore Limited to New York City which departs Chicago at 9:30 pm Central Time
Day 17
Lake Shore Limited arrives in New York City at 6:23 pm Eastern Time
Do you want to go to Florida or not, or will you be ready to fly back home to Europe?
Day 18
There are two Amtrak trains to Florida from New York City, Silver Star and Silver Meteor. Meteor has dining car services, the Star does not. Since you stated you had money to burn, lets take the Silver Meteor.
Catch the Silver Meteor to Miami which departs New York City at 3:15 pm Eastern Time
Day 19
Silver Meteor arrives in Miami at 6:39 pm Eastern Time. You will have to overnight in a hotel. Enjoy Miami Beach.
Day 20
Catch the Silver Meteor to New York City which departs Miami at 8:10 am Eastern Time
Day 21
Silver Meteor arrives in New York City at 7:23 pm Eastern Time.
The only large sections of track you saw twice were in opposite directions, (1) LA to Emeryville, and NYC to Miami. The only large section of track you saw three times was the NEC between NYC and D.C.
We still have a week of your month left which could have been spend anywhere along the way. You even have time to take the Cardinal west of New York City to Chicago and the City of New Orleans south of Chicago to New Orleans, and the Crescent north of New Orleans back to New York City. But the Cardinal is another three day a week train like the Sunset Limited, so if you wish to continue on lets start using letters for additional days.
Day A (Sunday-Wednesday-Friday)
Catch the Cardinal to Chicago which departs New York City at 6:45 am Eastern Time
Day B
Cardinal arrives in Chicago at 10:00 am Central Time
Catch the City of New Orleans to New Orleans which departs Chicago at 8:05 pm Central Time
Day C
City of New Orleans arrives in New Orleans at 3:47 pm Central Time. Enjoy another overnight in New Orleans.
Day D
Catch the Crescent to New York City which departs New Orleans at 7:00 am Central Time
Day E
Crescent arrives in New York City at 1:46 pm Eastern Time
That's an additional 5 days to the 21 days earlier on the train - totaling 26 days.
But the three days a week schedules of the Cardinal could add up to three more days to that total - depending upon the day of the week you started your trip.
Additionally, if you are not a gambler, and did not wish the one hour transfer in Portland, you could have taken the Coast Starlight all the way north to Seattle and caught the Empire Builder there the next day. You would have seen Puget Sound instead of the Columbia River Valley, and still arrived in Chicago on the same day. I still recommend the scenery of the Columbia River Valley over Puget Sound - because you see it for far much longer a time. Of course, visiting the Space Needle in Seattle while you overnight there might change your opinion. I recommended what I would do as you asked.