Price elasticity of demand in the point of economic equilibrium The function of demand is:\begin{align*} D(p) = 66-3p-p^2 \\\end{align*}
The function of supply is: \begin{align*}S(p) = 4p^2+8p-114\\
\end{align*}
The task is to find price elasticity of demand in the point of economic equilibrium.

I have found out that the equilibrium price is 5 and equilibrium demand is 26.

I also have a formula that states that $E = k * P/Q$
, where $P$ - equilibrium price, $Q$ - equilibrium demand and $k$ - coefficient  of $S(p)$ slope

How to find $k$ or are there another methods to solve this task?
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 $D'(p) P / D(p)$.