Unfortunately, long and cumbursome formulas completely obfuscate very simple nature of what's going on here. First of all, let me rescale the variables, so that $c=1$ and $v=1$. Now, Knudsen's law says that each time the particle hits the bottom, the vertical component $w=\cos\theta$ of the velocity is independently sampled from the distribution with the density $f(\xi)=2\xi$ on the interval $[0,1]$. The time until the particle hits the bottom again is $T=2/w$ (whatever happens on edges can be discarded as it has probability 0). Therefore, by the law of large numbers (which can be called many different names), the limit you are interested in is
$$
\frac{\mathbf E w}{\mathbf E T} = \frac{2/3}{4} = \frac16 \;.
$$