This seems too easy. We want the number of integers $ x $ which satisfy the equation $ (x^2-5x+5)^{x+5}=1 $. Thus we need either $ x^2 - 5x + 5 = 1 $, $ x + 5 = 0 $, or $ x^2 - 5x + 5 = -1 $ and $ x + 5 $ is even. Keep in mind that $ 0^0 $ is undefined. Our solutions to these three equations are $ x = -5, 1, 3, $ and $ 4 $. Thus, our answer is $ \boxed{4} $.