If you don’t have a desktop icon go to:
C:\Program Files\Adobe\Adobe Lightroom
Right-click on lightroom (the application) and choose Send to >> Desktop
If you already have a desktop icon for Adobe Lightroom try right-clicking on it and choosing “Run As Administrator”
An aside: Now, when I launch Creative Cloud, my installed apps have "Start Trial" instead. My membership doesn't expire until the end of July. (I started the trial for Lightroom, and it still won't launch.)
What package do you have?
From Creative Cloud try clicking on the settings icon (cogwheel) then sign out and sign-in again. See if that refreshes the apps to show LR installed.
I have Student and Teacher Edition, which expires on 31 July 2015.
I had already tried signing out and back in several times, with no success. After a reboot, I was informed that Creative Cloud was corrupted and needed to be reinstalled. I'm doing that right now, and I'll let you know if it works.
EDIT: Now I'm getting severely upset. I can't reinstall CC without uninstalling it first, and I can't uninstall it without uninstalling all installed CC programs. Problem is, I'm on a relatively slow Internet connection, and it'll take hours to redownload and reinstall those programs.
It should not be necessary to uninstall the apps.
I would suggest posting the problem over on the CC forum where there are more Adobe staff. They are able to check out your subscription at the back end.
I've reinstalled both CC and both apps.
Lightroom still won't launch. (And it's still showing the trial.)
Thanks to some help from the phone support at Adobe, I found the problem.
Due to a Windows bug, Windows 7 Service Pack 1 wasn't installed. (Windows Update should have installed it, but it wasn't installing anything.) After installing SP1, Lightroom started working.