Microwave penetration depth $lambda$ and surface resistance at 27 GHz are measured in high quality crystals of KOs$_2$O$_6$. Firm evidence for fully-gapped superconductivity is provided from $lambda(T)$. Below the second transition at $T_{rm p}sim 8$
K, the superfluid density shows a step-like change with a suppression of effective critical temperature $T_{rm c}$. Concurrently, the extracted quasiparticle scattering time shows a steep enhancement, indicating a strong coupling between the anomalous rattling motion of K ions and quasiparticles. The results imply that the rattling phonons help to enhance superconductivity, and that K sites freeze to an ordered state with long quasiparticle mean free path below $T_{rm p}$.
