Answer
$l=4$.
Work Step by Step
The smallest nonzero angle for a given $l$ occurs for $m=l$, when the angular momentum vector is almost aligned with the z-axis and has its maximum z-component. In that case, $L_z=l\hbar$ and $L=\sqrt{l(l+1)}\hbar$.
We are told:
$cos26.6^{\circ}=\frac{L_z}{L}=\frac{l}{\sqrt{l(l+1)}}$
Square the equation
$cos^2 26.6^{\circ}=\frac{l^2}{ l(l+1)}$
$ l(l+1)cos^2 26.6^{\circ}=l^2$
$l=\frac{ cos^2 26.6^{\circ}}{ 1- cos^2 26.6^{\circ}}=4$
