used on the way home to

Yes, it's fine, though punctuation would have made it clearer: ...she used, on the way home, to drop into my rooms, sometimes for an hour.

I thnk I would also have said "...on her way home..."

Separating the parts of "used to" sounds very awkward to me -- also, for some reason, "used to" seems too broad or vague to follow three different phrases each specifying the time of the action (after that; for more than a year; whenever Rosie came out with me). I would prefer "she would drop into my rooms on her way home." If you really want to say "used to," I would keep it together: "