The time-dependent $CP$ asymmetry in ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{K}^{\ensuremath{-}}$ decays is measured using $pp$ collision data, corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV. In a sample of 96 000 ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{K}^{\ensuremath{-}}$ decays, the $CP$-violating phase ${\ensuremath{\phi}}_{s}$ is measured, as well as the decay widths ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{L}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{H}}$ of the light and heavy mass eigenstates of the ${B}_{s}^{0}--{\overline{B}}_{s}^{0}$ system. The values obtained are ${\ensuremath{\phi}}_{s}=\ensuremath{-}0.058\ifmmode\pm\else\textpm\fi{}0.049\ifmmode\pm\else\textpm\fi{}0.006\text{ }\text{ }\mathrm{rad}$, ${\mathrm{\ensuremath{\Gamma}}}_{s}\ensuremath{\equiv}({\mathrm{\ensuremath{\Gamma}}}_{\mathrm{L}}+{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{H}})/2=0.6603\ifmmode\pm\else\textpm\fi{}0.0027\ifmmode\pm\else\textpm\fi{}0.0015\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$, and $\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Gamma}}}_{s}\ensuremath{\equiv}{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{L}}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{H}}=0.0805\ifmmode\pm\else\textpm\fi{}0.0091\ifmmode\pm\else\textpm\fi{}0.0032\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$, where the first uncertainty is statistical and the second, systematic. These are the most precise single measurements of those quantities to date. A combined analysis with ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays gives ${\ensuremath{\phi}}_{s}=\ensuremath{-}0.010\ifmmode\pm\else\textpm\fi{}0.039\text{ }\text{ }\mathrm{rad}$. All measurements are in agreement with the standard model predictions. For the first time, the phase ${\ensuremath{\phi}}_{s}$ is measured independently for each polarization state of the ${K}^{+}{K}^{\ensuremath{-}}$ system and shows no evidence for polarization dependence.