NP-Complete is not closed under kleene star
$L$ is $\mathrm{NP}$-complete, so, for any $X\in\mathrm{NP}$, there is a many-one reduction&nbsp;$f_X$ from $X$ to&nbsp;$L$.  Just modify $f_X$ so that it's a reduction to&nbsp;$L'$ instead. Note that $f_X$ already does the right thing unless you have some&nbsp;$w$ such that $f_X(w)\in\{0,1\}$.

Note that the same technique shows that $L\cup S$ is still $\mathrm{NP}$-complete for any finite $S\subseteq \Sigma^*$.