So in my continuous efforts, I realized that u = \sqrt{v_y^2 + v_x^2} so it's clear that v_y^2 = 2gH and, with a little work, that v_x^2 = \dfrac{gL^2}{8H}, but that's as far as I can get. Working backwards really doesn't help all that much. I've been trying to find where the L/4H comes from...