The wave scattering problem by a crack $\Gamma$ in $\mathbb{R}^2$ with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.