  In this work, we study the production of strange quarkoniums, the
$\phi(2170)$, also named $Y(2175)$, and the $\eta(2225)$, via a kaon induced
reaction on a proton target in an effective Lagrangian approach. The total and
differential cross sections of the reactions $K^{-}p\rightarrow \phi
(2170)\Lambda $ and $K^{-}p\rightarrow \eta (2225)\Lambda $ are calculated by
the Reggeized $t $-channel Born term under an assumption that the $\phi(2170)$
and $\eta(2225)$ are $\Lambda\bar{\Lambda}$ molecular states. At the center of
mass energies of about 4.2 GeV, the total cross section for the $\phi(2170)$
production is predicted to be about 1 $\mu $b. The numerical results indicate
that it is feasible to produce the $\phi (2170)$ via kaon beam scattering at
the best energy window near 4.2 GeV. The total cross section for the
$\eta(2225)$ production is smaller than that for the $\phi(2170)$ production
and it may reach an order of the magnitude of 0.1 $\mu$b. The differential
cross sections for both reactions at different center of mass energies are also
presented. It is found that the Reggeized $t$ channel gives a considerable
contribution at forward angles. As the energy increases, the contribution from
the $t$-channel almost concentrates at extreme forward angles. From these
theoretical predictions, the relevant experimental research is suggested, which
could provide important information to clarify the internal structure and
production mechanism of these two strange quarkoniums.
