In the immersed interface method, an object in a flow is formulated as a singular force, and jump conditions caused by the singular force are incorporated into numerical schemes to compute the flow. Previous development of the method considered only smooth objects. We here extend the method to handle non-smooth rigid objects with sharp corners in 2D incompressible viscous flows. We represent the boundary of an object as a polygonal curve moving through a fixed Cartesian grid. We compute necessary jump conditions to achieve boundary condition capturing on the object. We incorporate the jump conditions into finite difference schemes to solve the flow on the Cartesian grid. The accuracy, efficiency and robustness of our method are tested using canonical flow problems. The results demonstrate that the method has second-order accuracy for the velocity and first-order accuracy for the pressure in the infinity norm, and is extremely efficient and robust to simulate flows around non-smooth complex objects.