This is the assertion that I have in mind when I write "second isomorphism theorem for groups":
If $K$ and $N$ are subgroups of a group $G$, with $N$ normal in $G$, then $K /(N \cap K) \cong NK/N$.
I think that this assertion can be proven by considering the application $f \colon K \to NK/N$ determined by the rule $f(k) = Nk$, and showing that it is an epimorphism whose kernel is equal to $N \cap K$.
Do you know why it is that when some authors try to prove this result, instead of simply proceeding as above, they start off by considering the projection $\pi \colon G \to G/N$ and then restrict it to $K$? In my humble opinion, it is much more direct to analyze the application $f$ right from the start...
The map $K\to NK/N$ you defined in the first place is exactly the same map as written at the end of your post: the restriction of $G\to G/N$ to $K$, whose image is clearly $NK/N$.