  JWST observations have the potential to provide unprecedented constraints on
the history of reionization and the sources responsible for the ionizing
photons due to the detection of large populations of faint galaxies at $z\gg6$.
Modelling reionization requires knowing both the number of ionizing photons
that are produced by galaxies and the fraction of those photons that escape
into the intergalactic medium. Observational estimates of these values
generally rely on spectroscopy for which large samples with well-defined
selection functions remain limited. To overcome this challenge, we present an
implicit likelihood inference (ILI) pipeline trained on mock photometry to
predict the escaped ionizing luminosity of individual galaxies ($\dot{n}_{\rm
ion}$) based on photometric magnitudes and redshifts. Compared to traditional
SED-fitting methods, the new ILI pipeline is consistently more accurate and
significantly faster. We deploy the method on a sample of 4,559 high-redshift
galaxies from the JADES Deep survey, finding a gentle redshift evolution of
$\log_{10}(\dot{n}_{\rm ion}) = (0.08\pm0.01)z + (51.60\pm0.06)$, with
late-time values for $\dot{N}_{\rm ion}$ consistent with theoretical models and
observations. We measure the evolution of the volume-averaged ionized fraction
and optical depth to find that observed populations of star-forming galaxies
are capable of driving reionization to completion at $z\sim 5.3$ without the
need for AGN or other exotic sources. The $20\%$ of UV-brightest galaxies
($M_{\rm UV}<-18.5$) in our sample can reionize only $\sim30\%$ of the survey
volume, demonstrating that faint LyC emitters are crucial for reionization.
