Why does the Boltzmann constant appear in the entropy equation?

Why does the Boltzmann constant appear in the entropy equation?

It's because the conventional definition of entropy is tied to the concept of temperature,\[\mathrm{d}S = \int \frac{\delta q}{T}\] The concept of temperature pre-dates the scientific consensus that heat is microscopic kinetic energy. Therefore, when Anders Celsius introduced the scale that now bears his name, he did not understand the deep connection between temperature and kinetic energy, and could not use the concept of energy to define the concept of temperature. Instead, the centigrade degree became a new unit, which eventually served as the basis of the kelvin, the SI unit of temperature.

Entropy is therefore conventionally defined to have the SI units of joules per kelvin.

The statistical nature of entropy was not known until much later. When defined statistically, entropy is naturally dimensionless,\[S \sim -\sum_i p_i \log p_i = -\operatorname{tr} (\rho \log \rho)\] and in order to connect this to the conventional definition, which is based on temperature, a conversion factor has to be inserted:\[S := -k_B \sum_i p_i \log p_i = -k_B \operatorname{tr} (\rho \log \rho)\]
If statistical mechanics had been discovered before thermodynamics, then it is likely that there would be no Boltzmann constant at all, entropy would be dimensionless, and temperature would have the same units as energy,\[\frac{1}{T} = \frac{\partial S}{\partial E}\]

Why does Boltzmann's constant appear in the ideal gas law?

The ideal gas law in the form \[PV = Nk_B T\] links temperature (on the right hand side) with energy (on the left hand side). (Note that \(PV\) is the amount of work it takes to displace enough of the atmosphere to make space for a container of size \(V\) to hold the gas.) Therefore Boltzmann's constant is needed here.

What does the ideal gas law have to do with entropy?

They both link together temperature and energy.