A search is conducted for the electroweak pair production of a chargino and a neutralino $pp \rightarrow \tilde\chi^\pm_1 \tilde\chi^0_2$, where the chargino decays into the lightest neutralino and a $W$ boson, $\tilde\chi^\pm_1 \rightarrow \tilde\chi^0_1 W^{\pm}$, while the neutralino decays into the lightest neutralino and a Standard Model-like 125 GeV Higgs boson, $\tilde\chi^0_2 \rightarrow \tilde\chi^0_1 h$. Fully hadronic, semileptonic, diphoton, and multilepton (electrons, muons) final states with missing transverse momentum are considered in this search. Higgs bosons in the final state are identified by either two jets originating from bottom quarks ($h \rightarrow b\bar{b}$), two photons ($h \rightarrow \gamma\gamma$), or leptons from the decay modes $h \rightarrow WW$, $h \rightarrow ZZ$ or $h \rightarrow \tau \tau$. The analysis is based on 36.1 fb$^{-1}$ of $\sqrt{s} = 13$ TeV proton-proton collision data recorded by the ATLAS detector at the Large Hadron Collider. Observations are consistent with the Standard Model expectations, and 95% confidence-level limits of up to 680 GeV in $\tilde\chi^\pm_1/\tilde\chi^0_2$ mass are set in the context of a simplified supersymmetric model.