The distance of a point $(x, y)$ from the origin $(0, 0)$ is given by $\sqrt{x^2 + y^2}$. Given that the distances of $(-5, 5)$ and $(5, k)$ from the origin are equal, we have: $$ \begin{aligned} \sqrt{(-5)^2 + 5^2} &= \sqrt{5^2 + k^2} \\ (-5)^2 + 5^2 &= 5^2 + k^2 \\ 25 + 25 &= 25 + k^2 \\ 50 &= 25 + k^2 \\ k^2 &= 25 \\ k &= \pm 5 \end{aligned} $$ Since $5$ is one of the options, the correct value is $5$.