Here are some observations, too long for a comment:

1) Note that cuspidal irreducible representation are compactly induced

$\sigma = c-ind_K^G \tau = Ind_K^G \tau$

2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)

$Hom_G( c-ind_K^G \tau, \pi) = Hom_K( \tau, Res_K \pi)$

3) Silberger PGL(2) over the $p$ adics assect that $Res_K \pi$ is essentially $Ind_{B(o)}^{GL(2, o)} 1$ except for a finite dimensional part. I expect this to be true for $GL(n)$.

Hence classify the $\tau$ needed for $L_k$ ( I am not sure what your definition is here), and study how . $\Res_K Res_K \pi$ decomposes has been described for all irreducible cuspidal $\pi$.\pi$(Bushnell-Kutzko). In fact, I think that the supercuspidal representation form a semisimple category, so there the question might really reduce to something trivial, very much like for profinite groups. (profinite groups are actually exactly the compact locally profinite groups;) 1 Here are some observations, too long for a comment: 1) Note that cuspidal irreducible representation are compactly induced$\sigma = c-ind_K^G \tau = Ind_K^G \tau$2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)$Hom_G( c-ind_K^G \tau, \pi) = Hom_K( \tau, Res_K \pi)$3) Silberger PGL(2) over the$p$adics assect that$Res_K \pi$is essentially$Ind_{B(o)}^{GL(2, o)} 1$except for a finite dimensional part. I expect this to be true for$GL(n)$. Hence classify the$\tau$needed for$L_k$( I am not sure what your definition is here), and study how$\Res_K \pi$decomposes for all irreducible$\pi\$.