Buddy does not actually claim to regain anything at the end, but based on his feelings at the end and what we know of his memories, we could make the argument that he regains some of the value of their friendship, despite his feeling that he has completely lost a part of himself.
At the end of the story his friend has died and he says of finding out about her death,
"A message saying so merely confirms a piece of news some secret vein had already received, severing from me an irreplaceable part of myself, letting it loose like a kite on a broken string" (Capote).
Although he has lost his closest friend, it is obvious that in re-living his memories of her, he can look back on that relationship and see its worth. It is because he feels that a part of him has died that he fully understands how important she was to him and regains a sense of her as a person as well.