NFAs can be constructed for union, concatenation, and Kleene star

For union, first replace $N_L$ and $N_K$ by disjoint isomorphic copies. Add a fresh start state with $\epsilon$-transitions to the copied start states, keep all copied transitions, and take the union of the two copied accepting sets as the new accepting set. By [L1] and [L2], every accepting run chooses one copy at the start and then stays inside that copy, so a word is accepted exactly when one of the two original machines accepts it. Thus the new language is $L\cup K$.

For concatenation, take disjoint copies of $N_L$ and $N_K$, use the copied start state of $N_L$ as the new start state, add an $\epsilon$-transition from each copied accepting state of $N_L$ to the copied start state of $N_K$, and designate only the copied accepting states of $N_K$ as accepting. Then [L1] and [L3] show that a successful run is exactly a decomposition of the input as $uv$ with $u\in L$ and $v\in K$, so the new machine recognizes $LK$.

For star, add a fresh start state, make it the only accepting state, add an $\epsilon$-transition from it to the old start state of $N_L$, and add $\epsilon$-transitions from each old accepting state back to the fresh start state. By [L1] and [L3], every accepting run chooses finitely many factors from $L$, and every finite concatenation of words of $L$ is accepted. Therefore the language recognized is $L^{\ast }$.

Hence NFA's can be built for union, concatenation, and Kleene star.