Oh, let us have some fun with this...
First, you simply are not going to get the entire route straight and flat - not with engineering of the time, or close. That means there will be sections needing a different sort of engine than for the fast runs. You might switch out just the engine, or switch trains.
Still, there is a lot of fairly flat geography. Given enough(!) human determination, there could be some very long fast stretches.
Also you need to keep the rails very near exact in alignment. Given that ground can shift, this means a need for periodic re-alignment. To my mind this means a simple (hah!) mechanical gadget that runs along the rails, adjusting alignment. This is screws and wedges and a human watching an optical sight, so feasible for the time. (Yes, there would be cost.)
As others had noted, wider gauge is good for stability.
As other posters have noted, there is great advantage to an external power source. But electrifying the entire route seems improbable.
Mechanical gearboxes are going to be a problem - so use electric motors instead. There are motors and generators from that period still in service, so reliability could be good.
This means a steam turbine fitted to a generator, and the wheels driven by electric motors. Efficiency is a concern, but if we imagine rare-earth magnets available early (not a huge stretch), then this is feasible.
The Japanese bullet trains were built in the 1960s, so not huge gap from your time period. Would not be surprised if you could get equivalent 200mph speed.
Train would have a streamlined nose, much like present bullet trains. Powerplant would be a steam turbine, driving an electric generator.
When leaving station most power would come from electrified rails, and perhaps for tens of kilometers after. This reduces peak power requirements, and greatly stretches the on-board fuel. Might be electrified sections in more developed areas.
Likely want something near a closed-loop to reduce water carried.
Lots of air flowing past a fast train.
Back end of the engine would be huge radiator with aluminum fins
parallel to airflow - figure the last 10-20 meters of the engine sides
as a stack of fins.
(There are similar designs from the period, so not a stretch.)
Could such a thing have been built in that time. Maybe not, but also not so far from possible. :)