= Solution
The charged meson decays proceed through a virtual $W^+$:
$$
\pi^+(u\bar d)\to W^{+*}\to\mu^+\nu_\mu,\qquad
K^+(u\bar s)\to W^{+*}\to\mu^+\nu_\mu.
$$
The first <quark> vertex has CKM factor $V_{ud}$, while the second has $V_{us}$. In the simple Cabibbo description these are approximately $\cos\theta_C$ and $\sin\theta_C$, so the strange amplitude carries a suppression $\tan\theta_C$ relative to the nonstrange one, or $\tan^2\theta_C\simeq0.053$ in their corresponding squared weak-coupling factors.
For a charged pseudoscalar and a negligibly massive <neutrino>, the leading leptonic energy width has the form
$$
\Gamma(P^+\to\mu^+\nu_\mu)=
\frac{G_F^2f_P^2|V_{qq'}|^2}{8\pi}m_\mu^2m_P
\left(1-\frac{m_\mu^2}{m_P^2}\right)^2
$$
in natural units. The helicity factor $m_\mu^2$ is common to these two <muon> channels, while the threshold factor is much less favorable for the <pion>. Using the usual masses, its value is about $0.182$ for the <pion> and $0.910$ for the charged <kaon>: the <pion>'s threshold factor alone is only about one fifth as large. Including the explicit $m_P$ factor gives a kinematic ratio about $0.0564$. The <pseudoscalar decay constants> $f_P$ also matter, so <Lorentz-invariant phase space> and CKM factors alone do not determine the full lifetime ratio.<Leptonic Decays of Charged Pseudoscalar Mesons by Particle Data Group>[leading pseudoscalar leptonic-width formula and <pion>/<kaon> discussion]{ref}
For $K^+\to\pi^+\pi^0$, one tree topology has the spectator $u$ unchanged and $\bar s\to\bar u+W^{+*}$ followed by $W^{+*}\to u\bar d$. The spectator $u$ can combine with $\bar d$ to form $\pi^+$, and the $u\bar u$ component supplies $\pi^0$. This carries the strangeness-changing weak factor $V_{us}^*V_{ud}$. <Hadronization>, other <quark> rearrangements and strong final-state interactions determine the hadronic amplitude; the diagram is not a complete precision width calculation. The small <pion>–<muon> energy release suppresses the <pion> leptonic channel kinematically, whereas the strange weak vertex suppresses the <kaon> channels through <Cabibbo suppression>.
\Image[/past-exam-of-the-sub-atomic-physics-course-of-the-university-of-oxford/2016-pion-and-kaon-weak-decay-diagrams.png]
{title=Charged-pion and charged-kaon weak-decay tree diagrams}
{description=Original quark-level diagrams for the two leptonic channels and one spectator topology for the two-pion kaon channel. Antifermion arrows follow fermion flow, opposite their external momentum direction.}
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