Put him to work. Don't be mean or scoldy about it, just put his feet in motion. I wouldn't worry about him whinnying, just get her focused on you and responding to your requests. Notice that you almost never see a GP dressage horse, or a futurity reiner whinnying in the show pen. Focused horses don't whinny, so if you get the focus the whinnying will go away. If I had him in hand, I would put him on a circle around me at a trot and do some CA Lunging for Respect with her until he had a ear on me and was fairly keyed in to me. When my horses whinny under saddle to each other (usually the scenario is one is being ridden and the other is in the stall or paddock), I put them into a forward trot on a fairly loose rein and do fairly small circles, loops, figure eights until I have them willingly bending around my leg and following the cues from my seat and hands. When they get relaxed and focused again, I bring them down to a walk, let the reins out and pet and praise them.
The key is the changing of direction or speed. For an unfocused horse I like to keep speed pretty consistent at the trot. A trot satisfies any feeling they need to go, and is also harder for them to be lazy at. Changing direction gets them thinking about what you're telling them and listening to you. If you aren't familiar with CA's Lunging For Respect, it does incorporate lots of changes in direction.
Because you are causing his feet to move where you choose to have them move, while minimally moving your feet, you are the "boss mare." The trick is to not be "punishing" about it. Go at it with an attitude like "Oh! Poor fella, you seem to be bored. We can fix that, lets go to work for a few minutes."
Hope that helped! Good luck!