I've tried to explain the business case for getting more women into the company, but although they seem to go along with it, nothing is happening. I'm afraid that they are only paying lip service to the idea and aren't really committed.
A: I imagine these middle managers are mostly men. I'm not making glib assumptions about their misogyny, just that they may not be as comfortable interviewing and evaluating women as they are with other men. They're also probably used to running all-male teams with an all-male culture and simply don't know how they would cope with a female newcomer.
When trying to effect a change such as this, the trickiest bit is getting it going: the prospect is always more alarming than the reality.
So rather than trying to get general, theoretical commitment from all your middle managers, I suspect you'd be better off working closely with the one middle manager who seems least hostile to the idea. Your board, having bought the strategy, should also be helpful to you in being seen to encourage its implementation.
Aim to have one very able woman in place within, say, six months. Attitude changes may still be slow - but once it's seen to work in one group, resistance will start to dwindle. Whatever you do, don't persuade one of those middle managers to take on a woman against his will. If he wanted to show that it wasn't working, he easily could - and it would be grossly unfair on her.