In my view, this power rests with the House. The House issues the order and it's up to the House to interpret whether it is satisfied with the compliance with its order. It's the same with a committee's order, where the committee is given the opportunity to consider. There have been, again, in the precedent of the Afghan detainees, statements by the Speaker very clearly to that effect, that the House has a constitutional power to request this information.
At the same time, there is very strong encouragement to the House and committees to consider public policy imperatives when exercising those powers.