Deriviation of the "Twiddle Sum" property
The most straightforward way to see this is to note that for $k=mN$

$$W_N^{kn}=e^{j2\pi mnN/N}=e^{j2\pi mn}=1$$

So the sum for the case $k=mN$ is simply

$$\sum_{n=0}^{N-1}1=1+1+\ldots+1=N$$

Note that the solution using L'Hopital's rule is a bit dubious because for $k=mN$ the formula for the geometric series is not valid because the terms are all equal to one (note the $r\neq 1$ condition in the link above). So you arrive at an expression that is only valid for $k\neq mN$, and then you somehow massage it into a solution that also works for $k=mN$, but this is bad math, even for engineers.