Coupled-channel $\pi K$ and $\eta K$ scattering amplitudes are determined by studying the finite-volume energy spectra obtained from dynamical lattice QCD calculations. Using a large basis of interpolating operators, including both those resembling a $qq¯$ construction and those resembling a pair of mesons with relative momentum, a reliable excited-state spectrum can be obtained. Working at $m\pi =391\mathrm{MeV}$, we find a gradual increase in the $JP=0+$ $\pi K$ phase-shift which may be identified with a broad scalar resonance that couples strongly to $\pi K$ and weakly to $\eta K$. The low-energy behavior of this amplitude suggests a virtual bound-state that may be related to the $\kappa$ resonance. A bound state with $JP=1-$ is found very close to the $\pi K$ threshold energy, whose coupling to the $\pi K$ channel is compatible with that of the experimental $K\star (892)$. Evidence is found for a narrow resonance in $JP=2+$. Isospin--3/2 $\pi K$ scattering is also studied and non-resonant phase-shifts spanning the whole elastic scattering region are obtained.