I find the discussion of the Bohr atom by Michael Fowler very interesting.
The following is not meant as an answer, it is a description of Michael Fowler's approach.
It appears Michael Fowler has made use of historical documentation extensively. In two places in his discussion Micheal Fowler refers to written communication from Bohr to Rutherford. The discussion is quantitative; the concepts are discussed, and then expressed in formula.
The development of the ideas of Bohr is described as a two stage process.
The first stage was Bohr trying to find a viable model for a stable state of a multi-electron atom, on the assumption that the model must involve Planck's constant.
"Planck's constant plays a role in restricting allowed orbital changes in the oscillators in black body radiation—and these oscillators, although not very clearly understood, were of the same general size as atoms."
During this stage Bohr was guided by experimental data of atoms being bombarded with high velocity electrons, and giving of characteristic x-rays. The data appeared to corroboratre an expected correlation between electron speed (in the lowest energy state), and the total charge of the nucleus.
"In February 1913, Bohr was surprised to find out in a casual conversation with the spectroscopist H. R. Hansen that some patterns had been discerned in the apparent chaos of spectral lines. In particular, Hansen showed him Balmer's formula for hydrogen."
Balmer's formula had been discovered earlier, but Bohr was unaware of that.
Bohr said later:
"As soon as I saw Balmer's formula, the whole thing was immediately clear to me."
Bohr realized that the existence of a series in the spectrum shows that for Hydrogen, with its single electron there isn't just a single stable ground state, there is a sequence of stationary states.
As so often happens when a physicist does history of physics, probably this narrative has been much streamlined. No doubt there were many blind alleys, and only the succesful path is retained.
Still, we do have that quote from Bohr himself on his realization when he learned about the Balmer series. Clearly there was a big jump in his confidence level that he was on the right track.