The area formula for a rhombus is $ A = \frac 12 \cdot d_1 \cdot d_2 $, where $ d_1 $ and $ d_2 $ are the lengths of its two diagonals. The points $ (4,0) $ and $ (-4,0) $ are opposing vertices of the rhombus and both lie on the x-axis. Since the third point $ (0,K) $ lies on the y-axis, and the diagonals of a rhombus are perpendicular bisectors, it follows that the intersection of the diagonals must be at the origin. Thus, the last vertex is the point $ (0,-K) $. It follows that the diagonals have length $ 8 $ and $ 2K $, and the area is equal to $ 80 = \frac 12 \cdot 8 \cdot (2K) = 8K $. Thus, $ K = \frac{80}8 = \boxed{10} $.