The Quantum Story by Jim Baggott (2011) page 9 mentions that Planck apparently immediately recognized that Planck constant was a new fundamental physical constant, and could have potential applications in the definition of the system of units (TODO where was that published):

Planck wrote that the constants offered: 'the possibility of establishing units of length, mass, time and temperature which are independent of speciﬁc bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and nonhuman, constants which therefore can be called "fundamental physical units of measurement".'

This was a visionary insight, and was finally realized in the 2019 redefinition of the SI base units.

TODO how can it be derived from theoretical principles alone? There is one derivation at; en.wikipedia.org/wiki/Planck%27s_law#Derivation but it does not seem to mention the Schrödinger equation at all.

Video 1. Quantum Mechanics 2 - Photons by ViaScience (2012) Source. Contains a good explanation of how discretization + energy increases with frequency explains the black-body radiation experiment curve: you need more and more energy for small wavelengths, each time higher above the average energy available.

Wien approximation (1896)

Rayleigh-Jeans law (1900)

Derived from classical first principles, matches Planck's law for low frequencies, but diverges at higher frequencies.

Black-body radiation experiment

The Quantum Story by Jim Baggott (2011) page 10 mentions:
Early examples of such cavities included rather expensive closed cylinders made from porcelain and platinum.
and the footnote comments:
The study of cavity radiation was not just about establishing theoretical principles, however. It was also of interest to the German Bureau of Standards as a reference for rating electric lamps.
1859-60 Gustav Kirchhoff demonstrated that the ratio of emitted to absorbed energy depends only on the frequency of the radiation and the temperature inside the cavity
1896 Wien approximation seems to explain existing curves well
1900 expriments by Otto Lummer and Ernst Pringsheim show Wien approximation is bad for lower frequencies
1900-10-07 Heinrich Rubens visits Planck in Planck's villa in the Berlin suburb of Grünewald and informs him about new experimental he and Ferdinand Kurlbaum obtained, still showing that Wien approximation is bad
1900 Planck's law matches Lummer and Pringsheim's experiments well. Planck forced to make the "desperate" postulate that energy is exchanged in quantized lumps. Not clear that light itself is quantized however, he thinks it might be something to do with allowed vibration modes of the atoms of the cavity rather.
1900 Rayleigh-Jeans law derived from classical first principles matches Planck's law for low frequencies, but diverges at higher frequencies.