We present new measurements of the quasar luminosity function (LF) at z similar to 6 over an unprecedentedly wide range of the rest-frame ultraviolet luminosity M-1450 from -30 to -22 mag. This is the fifth in a series of publications from the Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs) project, which exploits the deep multiband imaging data produced by the Hyper Suprime-Cam Subaru Strategic Program survey. The LF was calculated with a complete sample of 110 quasars at 5.7 <= z <= 6.5, which includes 48 SHELLQs quasars discovered over 650 deg(2) and 63 brighter quasars discovered by the Sloan Digital Sky Survey and the Canada-France-Hawaii Quasar Survey (including one overlapping object). This is the largest sample of z similar to 6 quasars with a well-defined selection function constructed to date, which has allowed us to detect significant flattening of the LF at its faint end. A double power-law function fit to the sample yields a faint-end slope alpha = -1.23(-0.34)(+0.44), a bright-end slope beta = -2.73(-0.31)(+0.23), a break magnitude M*(1450) = -24.90(-0.90)(+0.75), and a characteristic space density Phi* = 10.9(-6.8)(+10.0) Gpc(-3) mag(-1). Integrating this best-fit model over the range -18 < M-1450 < -30 mag, quasars emit ionizing photons at the rate of <(n)over dot>(ion) = 10(48.8 +/- 0.1) s(-1) Mpc(-3) at z = 6.0. This is less than 10% of the critical rate necessary to keep the intergalactic medium ionized, which indicates that quasars are not a major contributor to cosmic reionization.