Measurements of charm mixing and $CP$ violation parameters from the decay-time-dependent ratio of ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ to ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ decay rates and the charge-conjugate ratio are reported. The analysis uses $\overline{B}\ensuremath{\rightarrow}{D}^{*+}{\ensuremath{\mu}}^{\ensuremath{-}}X$, and charge-conjugate decays, where ${D}^{*+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\pi}}^{+}$, and ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{\mp}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$. The $pp$ collision data are recorded by the LHCb experiment at center-of-mass energies $\sqrt{s}=7$ and 8 TeV, corresponding to an integrated luminosity of $3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The data are analyzed under three hypotheses: (i) mixing assuming $CP$ symmetry, (ii) mixing assuming no direct $CP$ violation in the Cabibbo-favored or doubly Cabibbo-suppressed decay amplitudes, and (iii) mixing allowing either direct $CP$ violation and/or $CP$ violation in the superpositions of flavor eigenstates defining the mass eigenstates. The data are also combined with those from a previous LHCb study of ${D}^{0}\ensuremath{\rightarrow}K\ensuremath{\pi}$ decays from a disjoint set of ${D}^{*+}$ candidates produced directly in $pp$ collisions. In all cases, the data are consistent with the hypothesis of $CP$ symmetry.