Is every subgroup of a normal subgroup normal?
The silly counterexample is this: if $H$ is not normal in $G$, then we have $$H \not\lhd G\quad G\lhd G$$ Indeed, this need not even be true if $K$ itself is normal in $H$. For example, in $S_4$, we have $$C_2 \lhd V_4\lhd S_4$$ but $C_2\not\lhd S_4$. (Here, $V_4 = \{(1), (12)(34),(13)(24),(14)(23)\}$ and $C_2 = \{(1), (12)(34)\}$)
The flaw in your argument is taking $ak = k_1 a$ where $k_1\in K$. The fact that $a\in G$ and $H \lhd G$ only allows you to assume that $k_1 \in H$.
G is a normal subgroup of itself, but it might have subgroups that are not normal.