In Ref. [1] we have presented the results of an exploratory lattice QCD computation of the long-distance contribution to the ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ decay amplitude. In the present paper we describe the details of this calculation, which includes the implementation of a number of novel techniques. The ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ decay amplitude is dominated by short-distance contributions which can be computed in perturbation theory with the only required nonperturbative input being the relatively well-known form factors of semileptonic kaon decays. The long-distance contributions, which are the target of this work, are expected to be of $O(5%)$ in the branching ratio. Our study demonstrates the feasibility of lattice QCD computations of the ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ decay amplitude, and in particular of the long-distance component. Though this calculation is performed on a small lattice ($1{6}^{3}\ifmmode\times\else\texttimes\fi{}32$) and at unphysical pion, kaon and charm quark masses, ${m}_{\ensuremath{\pi}}=420\text{ }\text{ }\mathrm{MeV}$, ${m}_{K}=563\text{ }\text{ }\mathrm{MeV}$ and ${m}_{c}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=863\text{ }\text{ }\mathrm{MeV}$, the techniques presented in this work can readily be applied to a future realistic calculation.