If by oxidation energy you mean of removing electron, it is popularly called ionization energy and for hydrogen atom it is R/n^2.
In Bohr you get the same energy as in Schrodinger: https://en.wikipedia.org/wiki/Bohr_model
Also in Bohr-Sommerfeld, which l=0 degeneration is basic model of Gryzinski, he has spin-orbit, Stark effect corrections.
Ten digits sounds like hyperfine correction: using also magnetic dipole moment of nucleus.
I haven't seen this kind of terms, numerical calculation was much more difficult in his times, but magnetic dipole of nucleus can be added there.
And generally this is not my field of research - I can help in recreating his work, but don't feel competent to do it alone.
ps. Talking about 10 digit energy accuracy - hyperfine level, the basic question to understand is why He3 and He4 behave so different - due to tiny difference of magnetic dipole moment of nucleus? Gryzinski comments it from perspective of his model in https://www.dropbox.com/s/r31tg6fu254jtwu/helium.pdf