jmc

Breaking the outer figure into two rectangles, we find that the total area of the shaded region plus unshaded region is $10\cdot 8+2\cdot 4=88$. Thus the area of the non-shaded region is $88-78=10$ square inches. This means that its remaining side length is 5 inches, and its perimeter is $2(2+5)=\boxed{14}$ inches.