Laser wakefield accelerators support extremely high accelerating fields, but dephasing, arising from the sub-luminal group velocity of the driving laser, limits both energy gain and laser-to-beam efficiency. Plasma density tailoring can mitigate dephasing, but determining the required profile in the nonlinear regime relevant to current experiments remains challenging. Here, we present a method to compute the density profile required for phase-locked laser wakefield acceleration using a lightweight closed-loop controller embedded within particle-in-cell simulations. Complementary analytical expressions for the nonlinear regime predict that phase-locking can increase electron energies by up to a factor of two, although quantitative accuracy requires a numerical approach to account for the spectral-temporal evolution of the laser pulse. Phase-locked simulations approximately double both the final energy and laser-to-beam efficiency relative to constant-density cases. These results demonstrate that density-tailored phase-locking is a practical route to efficient LWFA stages operating up to the depletion limit.