We present an atmospheric retrieval analysis of the Y0 brown dwarf WISE J035934.06$-$540154.6 using the low-resolution 0.96--12 $\mu$m JWST spectrum presented in \citet{Beiler_2023}. We obtain volume number mixing ratios of the major gas-phase absorbers (H$_2$O, CH$_4$, CO, CO$_2$, PH$_3$, and H$_2$S) that are 3--5$\times$ more precise than previous work that used HST spectra. We also find an order-of-magnitude improvement in the precision of the retrieved thermal profile, a direct result of the broad wavelength coverage of the JWST data. We used the retrieved thermal profile and surface gravity to generate a grid of chemical forward models with varying metallicity, (C/O)$_\textrm{atm}$, and strengths of vertical mixing as encapsulated by the eddy diffusion coefficient $K_\textrm{zz}$. Comparison of the retrieved abundances with this grid of models suggests that the deep atmosphere of WISE 0359$-$54 shows signs of vigorous vertical mixing with $K_\textrm{zz}=10^9$ [cm$^{2}$ s$^{-1}$]. To test the sensitivity of these results to our 5-knot spline thermal profile model, we performed a second retrieval using the \citet{Madhusudhan_2009} thermal profile model. While the results of the two retrievals generally agree well, we do find differences between the retrieved values of mass and volume number mixing ratio of H$_2$S with fractional differences of the median values of $-$0.64 and $-$0.10, respectively. In addition, the 5-knot thermal profile is consistently warmer at pressure between 1 and 70 bar. Nevertheless, our results underscore the power that the broad-wavelength infrared spectra obtainable with the James Webb Space Telescope have to characterize the atmospheres of cool brown dwarfs.
