I had to figure this out yesterday. First, as others have said, you have to view it from your kindle. Second, the lending library doesn't come up till after you are viewing the book on the full screen on the kindle.
I am still trying to decide if I want to keep the membership once the 'free' period is over. We don't order a lot throughout the year. The movie selection appears very week. I do like the 1 free book a month, but don't need it.
I still don't see the lending library option. I am viewing the book description on the kindle and see the "buy" option and a "send sample now" option at the right, but nothing about the lending library. Where should that button be? (BTW I also have downloaded multiple movies for free, so I know my "Prime membership" is associated with my account.) This is very frustrating.