There are two LR licensing models each with their own installer and program, one with a serial number, and one with a CC login.
You apparently have the serial-number version installed but you purchased a license for the CC-login one.
What you should do is uninstall the LR5 you have and reboot, then install LR from the CC Manager desktop app—the icon that looks like a lazy-8 or infinity symbol but is really two C’s facing each other. That’s the one that you’re paying $10/month for, not the serial-number one.
If you don’t yet have the CC Desktop app, then go to http://www.adobe.com, login with your Adobe ID that you used to purchase your CC license, and click the APPS / Desktop icon then click on the Download button under the LR icon..