Let E F G H, E F D C, and E H B C be three adjacent square faces of a cube, for which E C=8, and let A be the eighth vertex of the cube. Let I, J, and K be points on \overline{E F}, \overline{E H}, and \overline{E C}, respectively, so that E I=E . J=E K=2. A solid \mathcal{S} is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to \overline{A E}, and containing the edges \overline{I J}, \overline{J K}, and \overline{K I}. The surface area of \mathcal{S}. including the walls of the tunnel, is m+n \sqrt{p}, where m, n, and p are positive integers and p is not divisible by the square of any prime. Find m+n+p.