Kagawa's biggest problem is Rooney. Rooney, despite complaining about being asked to play out of position, has a free role in this United team, which explains why you see him in midfield (where Kagawa would be looking for the ball) asking for the ball (which he'll get) and spraying it out to the wings, so that he can run in to meet the crosses.
At the same time, the style that we play, is based exactly on just that, getting the ball, and spraying it out to the wingers, whether it's through Rooney in midfield, through Carrick, or via Rio / Vidic with their long balls to the wingers from the back... Yes, we occasionally play a through ball, but no where close to the intricate passing game which you will see from the likes of Barcelona, let alone Dortmund, the system which Kagawa was at his best with.
I think we can all agree that Kagawa really comes to life when he's in the no 10 role and there are two great examples of this season so far (one being the last game) where either, Rooney hasn't been on the pitch, or when Rooney has to play up top as the lone striker. Both of these occasions, he's able to show us glimpses of his former team's style, with flicks, turns and quick passes, essentially speeding up our build up, where there is a big difference to when Rooney plays, when he collects the ball, does a few passes and then looks for Valencia with a long square pass.
It's safe to say that as long as Rooney is at United, he's going to be starting and it's unlikely too that Moyes will ever tell him to stop dropping into midfield... which means unfortunately, that Kagawa, unless he reinvents himself as a winger, will continue to find it difficult to impose himself.