Hi.
Same problem.
I've had this Android app for almost a year, and (mostly) been happy with it. However, like you I've been unable to sign in for over a week now.
There's nothing wrong with my internet connection(s) - have used both mobile and WiFi. Have cleared the app cache, and even uninstalled and reinstalled. No joy.
My guess is that that the problem is 'out there (not on the phone), e.g. failing to access a Line 6 server (Line 6 maybe changed its configuration, or something). Hopefully they will wake up and fix whatever the problem is real soon, as the app isn't a lot of use without the internet connection. :-(
UPDATE: I raised a support ticket yesterday, and (while I've not yet had a reply) I see that it is working again, so presumably they have fixed whatever was wrong in their end. So, if you try again, you should be OK.