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Using $535\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B$-meson pairs collected by the Belle detector at the KEKB ${e}^{+}{e}^{\ensuremath{-}}$ collider, we measure branching fractions of $(7.16\ifmmode\pm\else\textpm\fi{}0.10(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.60(\mathrm{syst})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $(4.31\ifmmode\pm\else\textpm\fi{}0.20(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.50(\mathrm{syst}))\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\psi}}^{\ensuremath{'}}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. We perform amplitude analyses to determine the resonant structure of the ${K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ final state in ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\psi}}^{\ensuremath{'}}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and find that the ${K}_{1}(1270)$ is a prominent component of both decay modes. There is significant interference among the different intermediate states, which leads, in particular, to a striking distortion of the $\ensuremath{\rho}$ line shape due to the $\ensuremath{\omega}$. Based on the results of the fit to the ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ data, the relative decay fractions of the ${K}_{1}(1270)$ to $K\ensuremath{\rho}$, $K\ensuremath{\omega}$, and ${K}^{*}(892)\ensuremath{\pi}$ are consistent with previous measurements, but the decay fraction to ${K}_{0}^{*}(1430)$ is significantly smaller. Finally, by floating the mass and width of the ${K}_{1}(1270)$ in an additional fit of the ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ data, we measure a mass of $(1248.1\ifmmode\pm\else\textpm\fi{}3.3(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}1.4(\mathrm{syst}))\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ and a width of $(119.5\ifmmode\pm\else\textpm\fi{}5.2(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}6.7(\mathrm{syst}))\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ for the ${K}_{1}(1270)$.

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