Context. Models of stellar structure and evolution can be constrained by measuring accurate parameters of detached eclipsing binaries in open clusters. Multiple binary stars provide the means to determine helium abundances in these old stellar systems, and in turn, to improve estimates of their age. Aims: In the first paper of this series, we demonstrated how measurements of multiple eclipsing binaries in the old open cluster NGC 6791 sets tighter constraints on the properties of stellar models than has previously been possible, thereby potentially improving both the accuracy and precision of the cluster age. Here we add additional constraints and perform an extensive model comparison to determine the best estimates of the cluster age and helium content, employing as many observational constraints as possible. Methods: We improve our photometry and correct empirically for differential reddening effects. We then perform an extensive comparison of the new colour-magnitude diagrams (CMDs) and eclipsing binary measurements to Victoria and DSEP isochrones in order to estimate cluster parameters. We also reanalyse a spectrum of the star 2-17 to improve [Fe/H] constraints. Results: We find a best estimate of the age of ~8.3 Gyr for NGC 6791 while demonstrating that remaining age uncertainty is dominated by uncertainties in the CNO abundances. The helium mass fraction is well constrained at Y = 0.30±0.01 resulting in ΔY/ΔZ ~ 1.4 assuming that such a relation exists. During the analysis we firmly identify blue straggler stars, including the star 2-17, and find indications for the presence of their evolved counterparts. Our analysis supports the RGB mass-loss found from asteroseismology and we determine precisely the absolute mass of stars on the lower RGB, MRGB = 1.15±0.02 M&sun;. This will be an important consistency check for the detailed asteroseismology of cluster stars. Conclusions: Using multiple, detached eclipsing binaries for determining stellar cluster ages, it is now possible to constrain parameters of stellar models, notably the helium content, which were previously out of reach. By observing a suitable number of detached eclipsing binaries in several open clusters, it will be possible to calibrate the age-scale and the helium enrichment parameter Δ Y/Δ Z, and provide firm constraints that stellar models must reproduce. Based on observations carried out at the Nordic Optical Telescope at La Palma and ESO’s VLT/UVES ESO, Paranal, Chile (75.D-0206A, 77.D-0827A, 81.D-0091).Photometric data are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/543/A106