Figure covers at most 5 cells, so we need at least 4 steps to change color every cell. If we place figure 4 times so that center of figure lies in dark cell we consider that is possible to do that if $n=4$.

Furthermore, applying operation two times on same cells we get that is possible to get table with every cell is colored black in $n$ steps for every even $n \ge 4$.

We shall prove that for $n$ odd it is not possible to do that. Look at picture. Let $k$ be a difference between white and black cells in dark area in picture. Every figure placed at table covers odd number of dark cells, so after every step $k$ is changed for number $\equiv 2 \pmod{4}$. At the beginning $k=10$ at the and $k=-10$. From this it is clearly that we need even number of steps. So $n$ is every even number except 2.