Let \(EFGH\), \(EFDC\), and \(EHBC\) be three adjacent square faces of a cube, for which \(EC = 8\), and let \(A\) be the eighth vertex of the cube. Let \(I\), \(J\), and \(K\), be the points on \(\overline{EF}\), \(\overline{EH}\), and \(\overline{EC}\), respectively, so that \(EI = EJ = EK = 2\). A solid \(S\) is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to \(\overline{AE}\), and containing the edges, \(\overline{IJ}\), \(\overline{JK}\), and \(\overline{KI}\). The surface area of \(S\), 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 \(m + n + p\).

(第十九届AIME2 2001 第15题)