It seems the solution to the current crisis with signing into Xbox Live is to clear your system cache. Just did it myself and everything is up and running.
- Press the Guide button on your controller, go to Settings, and select System Settings.
- Select Storage or Memory.
- Highlight any storage device, and then press Y on your controller. (It doesn't matter which storage device you select; the cache will be cleared for all storage devices.)
- Select Clear System Cache.
- When prompted to confirm storage device maintenance, select Yes.