We present a fully coupled numerical method for simulating two-phase electrohydrodynamic flows, in particular, leaky dielectric fluids in an electric field. Within the leaky dielectric model, the electrical stress is treated as an interfacial force, and surface charges accumulate at the fluid interface. Consequently, the electrical force is incorporated into the incompressible Navier-Stokes equations using an extended Discontinuous Galerkin (XDG) method formulated within a hydrodynamic two-phase framework, in which the interface is represented by the zero-set of a level set function. The electrical potential is solved using the XDG formulation, which enforces both the continuity of the electrical potential and the continuity of the normal component of the electric flux density across the interface. In addition, the surface charge conservation equation is solved by employing additional trace-based methods. A series of numerical tests is conducted to illustrate the accuracy, stability, and applicability of the proposed method. Condition number analyses and convergence studies are performed to assess the numerical robustness and accuracy of the scheme. Furthermore, simulations with varying permittivity and conductivity ratios are presented to investigate droplet deformation dynamics in both two- and three-dimensional configurations. Droplet coalescence behavior under physically realistic conditions is also examined. The resulting sharp-interface model is solved in a fully coupled manner, enabling accurate high-order simulations. Interfacial discontinuities are sharply resolved without the need for additional reconstruction procedures, thereby providing a robust and efficient framework for the simulation of complex multiphase electrohydrodynamic phenomena.