  The decays of $B_s^0$ and $\overline{B}_s^0$ mesons into the $J/\psi K^+K^-$
final state are studied in the $K^+K^-$ mass region above the $\phi(1020)$
meson in order to determine the resonant substructure and measure the
$CP$-violating phase, $\phi_s$, the decay width, $\Gamma_s$, and the width
difference between light and heavy mass eigenstates, $\Delta\Gamma_s$. A
decay-time dependent amplitude analysis is employed. The data sample
corresponds to an integrated luminosity of $3{\rm fb}^{-1}$ produced in 7 and 8
Tev $pp$ collisions at the LHC, collected by the LHCb experiment. The
measurement determines $\phi_s = 119\pm107\pm34 {\rm \, mrad}$. A combination
with previous LHCb measurements using similar decays into the $J/\psi
\pi^+\pi^-$ and $J/\psi\phi(1020)$ final states gives $\phi_s=1\pm37 {\rm
\,mrad}$, consistent with the Standard Model prediction.
