Extracting precise cosmology from weak lensing surveys requires modelling the non-linear matter power spectrum, which is suppressed at small scales due to baryonic feedback processes. However, hydrodynamical galaxy formation simulations make widely varying predictions for the amplitude and extent of this effect. We use measurements of Dark Energy Survey Year 3 weak lensing (WL) and Atacama Cosmology Telescope DR5 kinematic Sunyaev-Zel'dovich (kSZ) to jointly constrain cosmological and astrophysical baryonic feedback parameters using a flexible analytical model, 'baryonification'. First, using WL only, we compare the $S_8$ constraints using baryonification to a simulation-calibrated halo model, a simulation-based emulator model, and the approach of discarding WL measurements on small angular scales. We find that model flexibility can shift the value of $S_8$ and degrade the uncertainty. The kSZ provides additional constraints on the astrophysical parameters, with the joint WL + kSZ analysis constraining $S_8=0.823^{+0.019}_{-0.020}$. We measure the suppression of the non-linear matter power spectrum using WL + kSZ and constrain a mean feedback scenario that is more extreme than the predictions from most hydrodynamical simulations. We constrain the baryon fractions and the gas mass fractions and find them to be generally lower than inferred from X-ray observations and simulation predictions. We conclude that the WL + kSZ measurements provide a new and complementary benchmark for building a coherent picture of the impact of gas around galaxies across observations.