Do you have a definition for a normal subgroup handy?
I'll give one that is reasonably easy to apply to your problem:
Definition: A subgroup H of G is normal iff .
Can you apply that definition for H=N and G=K?
Basically it means checking each combination of elements g and h.
Since it is trivial if either g or h is the identity, there are only 3 cases to check.
Alternatively, there is also a proposition that you may have:
Proposition: The kernel of a homomorphism from G to G' is a normal subgroup of G.
If you can find such a homomorphism this is less work.