  The $\overline{B_s^0} \rightarrow \chi_{c2} K^+ K^- $ decay mode is observed
and its branching fraction relative to the corresponding $\chi_{c1}$ decay
mode, in a $\pm 15 \textrm{MeV}/c^2$ window around the $\phi$ mass, is found to
be $\frac{\mathcal{B}(\overline{B_s^0} \rightarrow \chi_{c2} K^+ K^-) }{
\mathcal{B}(\overline{B_s^0} \rightarrow \chi_{c1} K^+ K^-)} = (17.1 \pm 3.1
\pm 0.4 \pm 0.9)\%,$ where the first uncertainty is statistical, the second
systematic and the third due to the knowledge of the branching fractions of
radiative $\chi_c$ decays. The decay mode $\overline{B_s^0} \rightarrow
\chi_{c1} K^+ K^- $ allows the $ B_s^0$ mass to be measured as $m(B_s^0) =
5366.83 \pm 0.25 \pm 0.27 \, \textrm{MeV}/c^2,$ where the first uncertainty is
statistical and the second systematic. A combination of this result with other
LHCb determinations of the $B_s^0$ mass is made.
