  For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and
$m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold
vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant
vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ in
$L_{\hat{\mathfrak{so}_m}}(1,0)^{\otimes 3}$ which can be realized as the
orbifold vertex operator subalgebra $L_{\hat{\mathfrak{sl}_2}}(2m,0)^{K}$ or
its extension. All the irreducible modules for
$L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and
$C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes
3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$ are classified and constructed
explicitly.
