We use momentum-transfer arguments to predict the friction factor f in two-dimensional turbulent soap film flows with rough boundaries (an analog of three-dimensional pipe flow) as a function of Reynolds number Re and roughness r , considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of f proportional, variant Re{-1/2} in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of Re{-1/4} . For large Re, f approximately r in the enstrophy-dominated case. We use conformal map techniques to perform direct numerical simulations that are in satisfactory agreement with theory and exhibit data collapse scaling of roughness-induced criticality, previously shown to arise in the three-dimensional pipe data of Nikuradse.