We consider the KPZ equation in space dimension 2 driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $ε$ and its strength is scaled as $\hatβ/ \sqrt{|\log ε|}$, then a transition occurs with explicit critical point $\hatβ_c = \sqrt{2π}$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $ε\downarrow 0$, for sufficiently small $\hatβ$. We prove here that the limit exists in the entire subcritical regime $\hatβ\in (0, \hatβ_c)$ and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension 2.