How did Mr. Dick actually facilitate the reconciliation between Dr. Strong and his wife Annie?
You haven't missed anything. You are completely right - all Mr. Dick actually does is use his position as a "simple-minded" individual to do and say what everyone else was not doing and saying so the Strongs can be reconciled. In Chapter 45 Mr. Dick approaches David Copperfield and asks him if David thinks he is simple minded. David agrees, which pleases Mr. Dick, and then he asks what the tension is between the Strongs. David explains, and then Mr. Dick decides to reconcile them because he is simple-minded and can therefore get away with behaviour that the "normal" people can't get away with.
So we can therefore assume that when he leads Annie into the room he has told her that now would be her opportunity to be reconciled to her husband, and we see the full reconciliation scene in Chapter 45.
I am glad someone is with me in the confusion here.
I suppose we can assume this..this is one of my favorite chapters so far (I have not finished the book.)
It seems with all that Strong/Annie nonsense in the previous chapters, everyone is jsut so obsessed with image and putting on airs, that nobody says anything and nobody gets anything done. Is that the thing about Dick? Is he supposed to be an anachronism or something, so he doesn't have to worry about the trappings of the day?
I suppose I assumed Dick not only led Annie into the room, but said something like "hey Doc is upset and I think this is why." Or, and this is a longshot...he could have told the Doc about the Will idea, to push Annie over the edge? I dunno...