The production of ultrabright electron bunches using ionization injection triggered by two transversely colliding laser pulses inside a beam-driven plasma wake is examined via three-dimensional particle-in-cell simulations. The relatively low intensity lasers are polarized along the wake axis and overlap with the wake for a very short time. The result is that the residual momentum of the ionized electrons in the transverse plane of the wake is reduced, and the injection is localized along the propagation axis of the wake. This minimizes both the initial thermal emittance and the emittance growth due to transverse phase mixing. Simulations show that ultrashort (~8 fs) high-current (0.4 kA) electron bunches with a normalized emittance of 8.5 and 6 nm in the two planes, respectively, and a brightness of 1.7×10(19) A rad(-2) m(-2) can be obtained for realistic parameters.