The sense of deep possessiveness toward one another is likewise a two-way street. While Kylo's feelings right now still ring loudly of concern, he can't deny the undercurrent of satisfaction that he's the one who was able to bring his grandfather here, and that he's the one who'd been left to watch over him, at least for now, even if both were largely matters of convenience.

That offer is yet another surprise. He knows what Rey's feelings toward Vader are--but he also knows Rey's tendency to want to help, so perhaps that isn't as much of a surprise as her having come here in the first place.

He won't force the issue if Vader doesn't want it--the biggest danger of his injury has already passed, at least--but he's quick to confirm with a grateful nod toward Rey once he says he'll accept it.