Timoshenko originally solved the equation with a $k$ factor. The $c^2$ factor came as a result of Timoshenko's analysis. The goal was to keep $k$ as a constant, which would make the equations easier. Unfortunately, $k$ is actually a function of frequency, which is unknown. This can be seen from a finite element or other elasticity analysis of the beam cross section with dynamic loading, and analyzing the shear stress in the cross sectional area.
In most low-frequency applications (like we see today), the effects from k can be ignored, so Timoshenko ignored these values. Instead, he focused on when the Euler equations began to deviate. For example, the Euler equations could not predict the phenomenon of a longitudinal wave. This occurs when a wave travels along the axis of a beam - like when stretching a spring:
The fundamental longitudinal wave frequency, therefore was considered highly important. Since this is a property that is based upon the material itself, as the $c$ in $c^2$ is the "speed of sound in the material", it made sense to find the value for $k$ such that Timoshenko's equations predicted this frequency. In other words, since $k$ is a function of frequency, when Timoshenko's equations were set to describe the longitudinal wave, the value for $k$ would correctly predict the first fundamental frequency. The net result is the $c^2$ term, which is accurate for this desired value of $k$. Other values of k can occur. For more information, there is a declassified resource from the Army Services I was able to consult - AD013061