"From what I can tell, sickbay is located on the short straight corridor that Kirk and Spock walk down after leaving the Transporter Room."
I believe you're right. The AG unit we see overhead behind Kelso appears to be the one connecting the wall with the "Environmental" ladder shaft with another (not seen in episode).
It's even more obvious in the shots with Dr. Piper leaving and Kirk entering. In the background you can clearly see the panel next to this room where Kirk and Spock disappear and Yeoman Doe exits in the alternate edit (turbolift or stairway) and illustrated in one of these two photos from The Making of Star Trek