Drag at the bed and along the lateral margins are the primary forces resisting flow in outlet glaciers. Simultaneously inferring these parameters is challenging since basal drag and ice viscosity are coupled in the momentum balance, which governs ice flow. We test the ability of adjoint-based inverse methods to infer the slipperiness coefficient in a power-law sliding law and the flow-rate parameter in the constitutive relation for ice using a regularization scheme that includes coefficients weighted by surface strain rates. Using synthetic data with spatial variations in basal drag and ice rheology comparable to those in West Antarctic Ice Streams, we show that this approach allows for more accurate inferences. We apply this method to Bindschadler and MacAyeal Ice Streams in West Antarctica. Our results show relatively soft ice in the shear margins and spatially varying basal drag, with an increase in drag with distance upstream of the grounding line punctuated by localized areas of relatively high drag. We interpret soft ice to reflect a combination of heating through viscous dissipation and changes in the crystalline structure. These results suggest that adjoint-based inverse methods can provide inferences of basal drag and ice rheology when regularization is informed by strain rates.