Unrefined as he is, The Question's true skill came from how adaptable he was in combat. He was a quick thinker, and was quite good at turning the tide of battle into a more favorable one.
Of course, this young woman was better than anyone he'd faced, save for Shiva or Richard.
Luckily for him, he was able to see the nerve strike coming, and move his left arm quicker than she could attack it. He lept backward for a moment, his eyeless face fixed on her mouthless one.
His mental process was quick. She could see through his feints, so he couldn't use feints. Perhaps...it was time to use the Old Hub City fighting style. And that, of course, was cheating.
After a few seconds of thought, The Question will almost attempt to bull rush Batgirl. As he's rushing towards her, in a fluid motion, he will reach down to grab some debris on the roof. His plan is obviously to throw some stuff in her eyes and then attack when she's blinded.