The task force gets a diary page from the second Kira and, with his inside knowledge, Light deduces that the meeting place is to be Aoyama - although he doesn't say this out loud.  L thinks otherwise but doesn't rule out any of the places listed.  Light does all he can to get there and to mask his identity as Kira but the second Kira finds him nonetheless.  Light is unaware that the Shinigami eyes cannot see the life span of another person who has a Death Note.