Well, there is always the possibility that Spock exits on Deck 7, the destination manually specified by Kirk (note his hand on the handle), while Kirk continues in the turbolift to his verbally specified destination of Deck 5. After all, when we next meet Kirk, he is in his quarters... This interpretation would also have Kirk escorting Spock basically all the way to Sickbay before backtracking to his own destination - a logical and kind action for the Captain, even if the dazed Spock then wanders about a bit after leaving the turbolift, confusing the issue.
The less likely alternative is that Kirk is taking Spock to his quarters on Deck 5, allowing him the dignity of combing his hair and washing that oddly enticing musky odor off him before he is to report to Sickbay. Naturally, Spock in his confused state wanders the two decks down on foot, forgoing the more logical paths. But where, then, is Kirk himself headed, when he ultimately ends up at his own quarters but does not step out on Deck 5? (Perhaps he resides on Deck 12 that day?)
Regarding the changing positioning of lifts close to Engineering, the concept of multiple roughly similar engine rooms rears its multiple roughly similar heads again...