The rare kaon decays K →π ν ν ¯ are strongly suppressed in the standard model and widely regarded as processes in which new phenomena, not predicted by the standard model, may be observed. Recognizing such new phenomena requires a precise standard model prediction for the branching ratio of K →π ν ν ¯ with controlled uncertainty for both short-distance and long-distance contributions. In this work we demonstrate the feasibility of lattice QCD calculation of the long-distance contribution to rare kaon decays with the emphasis on K+→π+ν ν ¯. Our methodology covers the calculation of both W -W and Z -exchange diagrams. We discuss the estimation of the power-law, finite-volume corrections and two methods to consistently combine the long-distance contribution determined by the lattice methods outlined here with the short-distance parts that can be reliably determined using perturbation theory. It is a subsequent work of our first methodology paper on K →π ℓ+ℓ-, where the focus was made on the γ -exchange diagrams.