If $N\lhd H×K$ then $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially

Remember: in general, $\,N\lhd G\Longrightarrow [G,N]\leq N$ , so in your case: $$N\lhd H\times K\Longleftrightarrow [H\times K:N]\leq N\Longrightarrow \,\,\text{in particular}\,\,[H:N]\,,\,[K:N]\leq N$$

where we identify $\,H\cong H\times 1\,\,,\,K\cong 1\times K\,$

Now suppose $\,N\,$ intersects both $\,H\,,\,K\,$ trivially, so $\,[H,N]\subset H\cap N =1\,$ and etc...can you take it from here?