If your current bike has a suspension then getting a bike without one could help some, especially on the hills. I wouldn't expect huge differences in time over a 9 mile distance by using a different bike. Maybe 5 minutes. 10 minutes at the outside. Could be more like 2 minutes or even 0 minutes.
As others have said, drop bars can make your life easier with headwinds. I think a lot people are more turned off by the riding position that many road bikes put you in rather than the drop bars themselves. Road bikes can be found that don't have the extreme drop from saddle to bars.
If you'd really prefer a hybrid there are lots of different bikes that fall into that category. I'd recommend what really amounts to a flat bar road bike.
Unless these are really steep hills I wouldn't worry about gearing too much. Even road bikes are available with triples or compact doubles which are adequate to get you up most anything (with a road).