did you try the Lightroom desktop menu at Help > Sign in?
I'm having the same problem this morning. The sign in screen appers but it seems to stall when I start typing in my address and password. Tried restarting LR and my iMac.
I can sign in to Adobe through my web browser without any problems though.
This is frustrating more than anything else, especially as it's not the first time this has happened.
Wish Adobe would sort out these annoying glitches with LR Mobile.
Could you try to re-sign-in via the Creative Cloud app /preferences. So quit Lr Desktop re-sign-in via the CC-app and relaunch Lr Desktop afterwards.
Let me know if that helps.
I've periodically suffered from this problem as well. The solution offered by Guido proved effective for me. I think my problem was that every time I launched LR from my task bar, as opposed to launching from within Creative Cloud, LR Mobile could not log in properly. I closed LR and re-launched via CC and LR Mobile worked as expected. Moral: Always launch apps from within CC.
Tried that several times, didn't work.