When I sign in to Lightroom Mobile it just hangs. Relaunched app, rebooted Macbook Pro 13" 2011. Internet connection fine, signing in to iOS apps fine (signed in and out to be sure), signing in to Adobe fine. Second day now.
App updated in Creative Cloud fine. Everything works except it won't sign me in.
Please post a screenshot of the Signing in window if it is showing any error.(blur your Email Id for privacy)
Also, follow the steps mentioned in Advanced solutions to connection errors with Adobe Creative Cloud and Creative Suite applications
and let us know if that helps.
I am also having this problem on a 2013 iMac OS X 10.11.6. When attempting to sign in via Lightroom desktop so I can sync new images to my phone, after entering email and password and hitting return, the sign in window just goes white and nothing happens.