Apply the Remainder Theorem.

According to the Remainder Theorem, if a polynomial $f(x)$ is divided by $(x-a)$, the remainder is $f(a)$. In this case, $f(x) = x^3 + kx^2 - 10x + 16$ is divided by $(x+1)$, so $a = -1$. The remainder is given as 18.
Therefore, we set $f(-1) = 18$.

$$f(-1) = (-1)^3 + k(-1)^2 - 10(-1) + 16 = 18$$

Calculate the powers and products.

$$-1 + k(1) + 10 + 16 = 18$$ $$-1 + k + 10 + 16 = 18$$

Combine the constant terms on the left side.

$$k + 25 = 18$$

Subtract 25 from both sides to isolate $k$.

$$k = 18 - 25$$ $$k = -7$$

The value of $k$ is $\boxed{-7}$.