Why can entropy be measured in joules per kelvin?
The definition of entropy in 'Classical Thermodynamics' is $$\Delta S = \int\frac{\delta Q}{T}$$

The quantity on the left is the entropy change associated with a physical process. $\delta Q$ is an inexact differential, made exact by an integrating factor $\frac{1}{T}$ This notion was devolped by Kelvin and Cartheodory (I believe). This comes out of the analysis of certain kinds of differential equations called Pfaffian forms. 

One notes the dimensions ("units") of this quantity, and observers them to be $\mathrm{J\ K^{-1}}$

It was only later, that the statistical definition of entropy (developed by Boltzman) came to be (and was found to be equivalent to the thermodynamic definition).

$$S = -k_B \sum_{i}p_i\ln(p_i)$$

In this framework entropy is a logarithmic measure of the number of states with significant probability of being occupied. In this expression, the the quantity is changed from dimensionless to dimensionful, by the Boltzmann Constat, which has the units, (you guesed it)  $\mathrm{J\ K^{-1}}$

The measure of "disorder" is in the term $\sum_{i}p_i\ln(p_i)$, which has no units. The units are appended to it by the Boltzmann Constant.