The Dalitz plot distribution of ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays is studied using a data sample corresponding to $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. The data are described by an amplitude model that contains contributions from intermediate ${K}^{*}(892{)}^{0}$, ${K}^{*}(1410{)}^{0}$, ${K}_{2}^{*}(1430{)}^{0}$ and ${D}_{2}^{*}(2460{)}^{\ensuremath{-}}$ resonances. The model also contains components to describe broad structures, including the ${K}_{0}^{*}(1430{)}^{0}$ and ${D}_{0}^{*}(2400{)}^{\ensuremath{-}}$ resonances, in the $K\ensuremath{\pi}$ S-wave and the $D\ensuremath{\pi}$ S- and P-waves. The masses and widths of the ${D}_{0}^{*}(2400{)}^{\ensuremath{-}}$ and ${D}_{2}^{*}(2460{)}^{\ensuremath{-}}$ resonances are measured, as are the complex amplitudes and fit fractions for all components included in the amplitude model. The model obtained will be an integral part of a future determination of the angle $\ensuremath{\gamma}$ of the Cabibbo-Kobayashi-Maskawa quark mixing matrix using ${B}^{0}\ensuremath{\rightarrow}D{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays.