 Direct-product structure of a finite minimal normal subgroup

Choose a minimal nontrivial normal subgroup $L$ of $K$. Its conjugates under the ambient group are minimal normal subgroups of $K$; distinct ones intersect trivially and commute. Choose a maximal family whose product is direct. Any omitted conjugate either lies in that product or meets it trivially and could be adjoined, so all conjugates lie in it. Their product is ambient-normal and therefore is $K$. A normal subgroup of one factor is then normal in $K$, since the other factors commute with it; minimality makes each factor simple. All factors are isomorphic because they are conjugates of $L$. In the abelian case they are cyclic of one common prime order.