  Faint $z\sim5$ quasars with $M_{1450}\sim-23$ mag are known to be the
potentially important contributors to the ultraviolet ionizing background in
the post-reionization era. However, their number density has not been well
determined, making it difficult to assess their role in the early ionization of
the intergalactic medium (IGM). In this work, we present the updated results of
our $z\sim5$ quasar survey using the Infrared Medium-deep Survey (IMS), a
near-infrared imaging survey covering an area of 85 deg$^{2}$. From our
spectroscopic observations with the Gemini Multi-Object Spectrograph (GMOS) on
the Gemini-South 8 m Telescope, we discovered eight new quasars at $z\sim5$
with $-26.1\leq M_{1450} \leq -23.3$. Combining our IMS faint quasars
($M_{1450}>-27$ mag) with the brighter Sloan Digital Sky Survey (SDSS) quasars
($M_{1450}<-27$ mag), we derive the $z\sim5$ quasar luminosity function (QLF)
without any fixed parameters down to the magnitude limit of $M_{1450}=-23$ mag.
We find that the faint-end slope of the QLF is very flat
($\alpha=-1.2^{+1.4}_{-0.6}$), with a characteristic luminosity of
$M^{*}_{1450}=-25.8^{+1.4}_{-1.1}$ mag. The number density of $z\sim5$ quasars
from the QLF gives an ionizing emissivity at 912 $\unicode{x212B}$ of
$\epsilon_{912}=(3.7$--$7.1)\times10^{23}$ erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$
and an ionizing photon density of $\dot{n}_{\rm ion}=(3.0$--$5.7)\times10^{49}$
Mpc$^{-3}$ s$^{-1}$. These results imply that quasars are responsible for only
10-20% (up to 50% even in the extreme case) of the photons required to
completely ionize the IGM at $z\sim5$, disfavoring the idea that quasars alone
could have ionized the IGM at $z\sim5$.
