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)