Question about direct factor and direct product.

Question

If $G$ is a group, $G=KK'$, $K=HH'$, where $K,K'$are normal subgroups of $G$ such that $K \cap K'=\left \{e\right \}$ and $H,H'$ are normal subgroups of $K$ such that $H \cap H'=\left \{e\right \}$. Prove: $H$ is normal in $G$. It's from this question: If $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$, then $H$ is a direct factor of $G$. I think it's equavelent to prove the above. So take any element of $G=KK'=K'K$ and any element of $H$, say $g=k'k \in G$ and $h \in H$, $ghg^{-1}=k'khk^{-1}k'^{-1}$ ,I know $khk^{-1} \in H$, then what should I do next to prove $k'khk^{-1}k'^{-1} \in H$?