A dimensional analysis path to $h$ and the Bohr atom structure

Traditionally, the Planck constant $h$ makes its debut appearance in quantum physics textbooks in the context of the blackbody radiation law and subsequently as a fundamental ingredient of the physics of the photoelectric effect and the Bohr atom. In this paper we consider an alternative timeline path (which could have taken place several years before the proposal of the Bohr atom) where a classical physics hydrogen atom is studied with dimensional analysis techniques in combination with empirical laws of blackbody radiation. The outcome of this ``classical physicist's'' approach is the identification of the correct fundamental Planck constant and the reconstruction of the energy and size scales of the Bohr atom.