We report measurements of charm-mixing parameters based on the decay-time-dependent ratio of ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ to ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ rates. The analysis uses a data sample of proton-proton collisions corresponding to an integrated luminosity of $5.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ recorded by the LHCb experiment from 2011 through 2016. Assuming charge-parity ($CP$) symmetry, the mixing parameters are determined to be ${x}^{\ensuremath{'}2}=(3.9\ifmmode\pm\else\textpm\fi{}2.7)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, ${y}^{\ensuremath{'}}=(5.28\ifmmode\pm\else\textpm\fi{}0.52)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and ${R}_{D}=(3.454\ifmmode\pm\else\textpm\fi{}0.031)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. Without this assumption, the measurement is performed separately for ${D}^{0}$ and ${\overline{D}}^{0}$ mesons, yielding a direct $CP$-violating asymmetry ${A}_{D}=\phantom{\rule{0ex}{0ex}}(\ensuremath{-}0.1\ifmmode\pm\else\textpm\fi{}9.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and magnitude of the ratio of mixing parameters $1.00<|q/p|<1.35$ at the 68.3% confidence level. All results include statistical and systematic uncertainties and improve significantly upon previous single-measurement determinations. No evidence for $CP$ violation in charm mixing is observed.