The interaction between $\Lambda$ baryons and kaons/antikaons is a crucial part of the strangeness $S=0$ ($\Lambda-\mathrm{K}$) and $S=-2$ ($\Lambda$-$\mathrm{\overline{K}}$) sector of the meson-baryon interaction at low energies and in the understanding of states such as the $\Xi(1620)$, whose nature and properties are still rather unknown. Experimental data on $\Lambda$-K and $\Lambda$-$\mathrm{\overline{K}}$ systems are rather scarce, leading to large uncertainties and tension between the available theoretical predictions. Recently, measurements of two-particle correlations in small colliding systems such as p-p collisions at LHC provided a significant improvement in the knowledge of several hadron-hadron interactions, in particular when strange baryons and mesons are involved. Correlation functions of $\Lambda - $K$^+\oplus \overline{\Lambda} - $K$^-$ and $\Lambda - $K$^-\oplus \overline{\Lambda} - $K$^+$ pairs were measured in Pb-Pb collisions and provided for the first time experimental constraints on the scattering parameters of the underlying interactions. In this letter we extend the measurements to small colliding systems by presenting $\Lambda - $K$^+\oplus \overline{\Lambda} - $K$^-$ and $\Lambda - $K$^-\oplus \overline{\Lambda} - $K$^+$ correlations obtained in the high-multiplicity triggered sample in p-p collisions at $\sqrt{s}=13$ TeV recorded by ALICE. The interaction for both pairs is modeled using the Lednicky-Lyuboshits analytical formula and the corresponding scattering parameters $Ref_0$, $Imf_0$ and $d_0$ are extracted by comparing the theoretical correlation to the data. A repulsive interaction for the $\Lambda - $K$^+\oplus \overline{\Lambda} - $K$^-$ system is found, which is in agreement with previous Pb-Pb results. The non-negligible $Imf_0$ indicates a significant coupling to inelastic channels, such as $\pi$N and $\Sigma$K. The $\Lambda - $K$^+\oplus \overline{\Lambda} - $K$^-$ correlations show the presence of several structures at relative momenta $k^\ast$ above 200 MeV$/c$, compatible with the $\Omega$, $\Xi(1690)$ and $\Xi(1820)$ resonances decaying into $\Lambda-\mathrm{K}^-$ pairs. The low $k^\ast$ region in the $\Lambda - $K$^-\oplus \overline{\Lambda} - $K$^+$ also exhibits the presence of the $\Xi(1620)$ state, expected to strongly couple to the measured pair. The $\Lambda - $K$^-\oplus \overline{\Lambda} - $K$^+$ interaction is hence modeled assuming a resonant contribution, through the formation of the $\Xi(1620)$ state, and a non-resonant one, which dominates the intermediate $k^\ast$ region. The extracted non-resonant scattering parameters are compatible with the Pb-Pb measurements and the extracted pole properties of the $\Xi(1620)$ agree with the recent Belle measurement of $\Xi(1620)\rightarrow\Xi\pi$.  The presented data allow to access the $\Lambda - $K$^+\oplus \overline{\Lambda} - $K$^-$ and $\Lambda - $K$^-\oplus \overline{\Lambda} - $K$^+$ interaction with an unprecedented precision and the first experimental observation of the $\Xi(1620)$ decaying into the $\Lambda-\mathrm{K}^-$ channel.