I imagine you've figured it out by now or have given up but for anyone looking, you might want to try restarting Xbox Live Auth Manager in Windows Services
Ctrl+Shift+Esc to open task manager
Go to more details
Select Services, select open services at the bottom
Then scroll down until you see Xbox Live Auth Manager
If you can't simply right click and start it, then open properties and change startup type to automatic
Click apply, then start it and it should allow you to sign in now.