Elasticity when the demand function is given
The definition of elasticity of demand with respect to price is: $\varepsilon_{q,p} = \frac{dq}{dp} \cdot \frac{p}{q}$. So in your demand function we have:

$$q = kp^{-\epsilon}$$
$$\frac{dq}{dp} = -\epsilon kp^{-\epsilon - 1}$$
$$\varepsilon_{q,p} = \frac{dq}{dp}\cdot \frac{p}{q}  = -\epsilon kp^{-\epsilon - 1} \cdot \frac{p}{kp^{-\epsilon}} = -\epsilon$$ 

And we have the answer you're looking for.