Price elasticity of demand in the point of economic equilibrium
Your formula is overly simplified.

The elasticity is supposed to be "how much does supply change if the price changes". Now, the natural way of looking at this is $S'(p)$, the derivative of $S$. However, the units of that are hard to make use of. Hence, we normalize $S'(p)$ by multiplying it with $P/S(p)$, hence getting the "percentage response of $S$ to a percent change in $P$".

Long story short, your $k$ should satisfy 

$$ k = S'(p)$$

Note that this is the price elasticity of supply. For the price elasticity of demand, you would instead use $D'(p) P / D(p)$.