(Note: I have treated the question as a classical covering problem, while OP apparently intended that the occupied squares need to be attacked as well. I'm leaving the answer up anyway, since this interpretation yields an interesting puzzle too.)
Here's the biggest one I got:
It took surprisingly long to fiddle the placements so that everything fit, so my guess is that the next size is not possible anymore.
* Rooks work at optimum efficiency in the corners too, so use them to shrink the board
* The diagonal pieces want to be near the centre of the board to cover the maximum number of squares.
* Use the rest of the pieces to plug any holes left over.
* Try again.
* And again.
Original answer below.
Apart from finding suitable software, it's pretty simple to cover a
14 x 14
board with the pieces. Here's how:
Seeing how the queens are suboptimally placed and there's a whole unused piece, this is almost certainly not the maximum. I'll try to cover the next bigger board and update soonish.