In this work, we study the localized $CP$ violation in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}\ensuremath{\sigma}(600)$ decays by employing the quasi-two-body QCD factorization approach. Both the resonance and the nonresonance contributions are studied for the ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay. The resonance contributions include those not only from $[\ensuremath{\pi}\ensuremath{\pi}]$ channels including $\ensuremath{\sigma}(600)$, ${\ensuremath{\rho}}^{0}(770)$ and $\ensuremath{\omega}(782)$ but also from $[K\ensuremath{\pi}]$ channels including ${K}_{0}^{*}(700)(\ensuremath{\kappa})$, ${K}^{*}(892)$, ${K}_{0}^{*}(1430)$, ${K}^{*}(1410)$, ${K}^{*}(1680)$ and ${K}_{2}^{*}(1430)$. By fitting the four experimental data ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=0.678\ifmmode\pm\else\textpm\fi{}0.078\ifmmode\pm\else\textpm\fi{}0.0323\ifmmode\pm\else\textpm\fi{}0.007$ for ${m}_{{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}}^{2}<15\text{ }\text{ }{\mathrm{GeV}}^{2}$ and $0.08<{m}_{{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}}^{2}<0.66\text{ }\text{ }{\mathrm{GeV}}^{2}$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{0}^{*}(1430){\ensuremath{\pi}}^{\ensuremath{-}})=0.061\ifmmode\pm\else\textpm\fi{}0.032$, $\mathcal{B}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{0}^{*}(1430){\ensuremath{\pi}}^{\ensuremath{-}})=({39}_{\ensuremath{-}5}^{+6})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $\mathcal{B}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\sigma}(600){\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}\phantom{\rule{0ex}{0ex}}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})<4.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, we get the end-point divergence parameters in our model, ${\ensuremath{\phi}}_{S}\ensuremath{\in}[1.77,2.25]$ and ${\ensuremath{\rho}}_{S}\ensuremath{\in}[2.39,4.02]$. Using these results for ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\phi}}_{S}$, we predict that the $CP$ asymmetry parameter ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}\ensuremath{\in}[\ensuremath{-}0.34,\ensuremath{-}0.11]$ and the branching fraction $\mathcal{B}\ensuremath{\in}[6.53,17.52]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ for the ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}\ensuremath{\sigma}(600)$ decay. In addition, we also analyze contributions to the localized $CP$ asymmetry ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ from $[\ensuremath{\pi}\ensuremath{\pi}]$, $[K\ensuremath{\pi}]$ channel resonances and nonresonance individually, which are found to be ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}[{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}]\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=0.509\ifmmode\pm\else\textpm\fi{}0.042$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}[{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}]\ensuremath{\pi}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=0.174\ifmmode\pm\else\textpm\fi{}0.025$ and ${{\mathcal{A}}_{\mathcal{C}\mathcal{P}}}^{\mathrm{NR}}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=0.061\ifmmode\pm\else\textpm\fi{}0.0042$, respectively. Comparing these results, we can see that the localized $CP$ asymmetry in the ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay is mainly induced by the $[\ensuremath{\pi}\ensuremath{\pi}]$ channel resonances while contributions from the $[K\ensuremath{\pi}]$ channel resonances and nonresonance are both very small.