Based on the \begin{document}$ H^2 $\end{document} existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle \begin{document}$ \omega\in(0,\pi/2) $\end{document} . This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when \begin{document}$ \omega\in(0,\pi) $\end{document} .