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We present results on $B\ensuremath{\rightarrow}J/\ensuremath{\psi}K\ensuremath{\pi}$ decays using ${e}^{+}{e}^{\ensuremath{-}}$annihilation data collected with the BABAR detector at the $\ensuremath{\Upsilon}(4S)$ resonance. The detector is located at the PEP-II asymmetric-energy storage ring facility at the Stanford Linear Accelerator Center. Using approximately $88\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs, we measure the decay amplitudes for the flavor eigenmodes and …