Chirality purification of single-walled carbon nanotubes (SWCNTs) is desirable for applications in many fields, but general utility is currently hampered by low throughput. We discovered a method to obtain single-chirality SWCNT enrichment by the aqueous two-phase extraction (ATPE) method in a single step. To achieve appropriate resolution, a biphasic system of non-ionic tri-block copolymer surfactant is varied with an ionic surfactant. A nearly-monochiral fraction of SWCNTs can then be harvested from the top phase. We also found, via high-throughput, near-infrared excitation-emission photoluminescence spectroscopy, that the parameter space of ATPE can be mapped to probe the mechanics of the separation process. Finally, we found that optimized conditions can be used for sorting of SWCNTs wrapped with ssDNA as well. Elimination of the need for surfactant exchange and simplicity of the separation process make the approach promising for high-yield generation of purified single-chirality SWCNT preparations.