He's a central midfield player capable of operating in the "#10 role" and out wide if need be (but there shouldn't be any need for it were he to come here).
I know Fergie's gone now but we've never had an "enforcer" per say. I mean sure you could take a look at Keane and tell me I'm wrong, but Keane was so much more than just an enforcer it shouldn't even be a debate. Besides, in this case Kagawa would be operating behind the striker, more or less being the "tip" of the midfield where-as Carrick and Thiago would be played as "CMs". And yes, Carrick isn't a "destroyer" like Tiote, but you don't need to be one if you're as good as he is at reading the game. I'm not saying you won't find a more sound midfielder in terms of shielding the defense, but he doesn't need to chase the ball as much because he knows how to position himself defensively (most notably closing down passing lanes and intercepting passes). I don't know where the idea of those 3 "not working" is coming from. We've played Kagawa behind the striker with Carrick in midfield more than once, the only difference now would be that Carrick would have a partner with more quality.
Not to mention we might just end up with Kagawa playing on the left while drifting in with Rooney staying and playing that role.
I for one don't find any problem with either of the two set-ups:
Carrick------Thiago
-------Kagawa-----
Carrick------Thiago
-------Rooney-----