Valve basically sends a verification code to the primary account email address that you need to enter before you can complete the sign in.
Chance is, if the former did not work, this won't work as well.
If you have a second PC, you can try the Steam connection there, just to test it out.
If it is not working as well, it could indicate a issue either with your Internet connection, or with Steam servers.
Problem: Steam showing updates in clients, even when it's fully updated on server.
Cause: Steam requires its service to be running when client boots, if the steam was installed in the Game disk in server then the steam service is not in auto start in client.