  We fill up the missing piece in our own re-analysis of the long-distance
electromagnetic radiative corrections to semileptonic kaon decays by performing
a rigorous study in the $K\rightarrow\pi\mu^+\nu_\mu(\gamma)$ channels. With
appropriate experimental and lattice inputs, we achieve a precision level of
$10^{-4}$ in these channels. This is comparable to our previous analysis in the
$K\rightarrow\pi e^+\nu_e(\gamma)$ channels. With this new result, we present
an updated global analysis to extract the Standard Model parameter $|V_{us}|$
from semileptonic kaon decays. We obtain $|V_{us}|=0.22308(55)$ and
$0.22356(73)$, using the lattice average of the $K^0\rightarrow\pi^-$
transition form factor at $N_f=2+1+1$ and $N_f=2+1$, respectively.
