We present cross sections for the reaction e^{+}e^{-}→K_{S}^{0}K_{L}^{0} at center-of-mass energies ranging from 3.51 to 4.95 GeV using data samples collected in the BESIII experiment, corresponding to a total integrated luminosity of 26.5 fb^{-1}. The ratio of neutral-to-charged kaon form factors at large momentum transfers (12<Q^{2}<25 GeV^{2}) is determined to be 0.21±0.01, which indicates a small but significant effect of flavor-SU(3) breaking in the kaon wave function, and, consequently, excludes the possibility that flavor-SU(3) breaking is the primary reason for the strong experimental violation of the pQCD prediction |F(π^{±})|/|F(K^{±})|=f_{π}^{2}/f_{K}^{2}, where F(π^{±}) and F(K^{±}) are the form factors, and f_{π} and f_{K} are the decay constants of charged pions and kaons, respectively. We also observe a significant signal for the charmless decay ψ(3770)→K_{S}^{0}K_{L}^{0} for the first time. Within a 1σ contour of the likelihood value, the branching fraction for ψ(3770)→K_{S}^{0}K_{L}^{0} is determined to be B=(2.63_{-1.59}^{+1.40})×10^{-5}, and the relative phase between the continuum and ψ(3770) amplitudes is ϕ=(-0.39_{-0.10}^{+0.05})π. The branching fraction is in good agreement with the S- and D-wave charmonia mixing scheme proposed in the interpretation of the "ρπ puzzle" between J/ψ and ψ(3686) decays.