Hi, thought I'd add a quick message here as I seem to have fixed the login problem (at least for me) that a lot of people are having after the recent update.
In settings > system > storage > hard drive > gamer profiles, when I checked my profile it said it was corrupted. Once I deleted this in full, and then downloaded it again from the internet, I'm having no problems logging in to xbox live.
Do have to say it was a waste of my morning having to deal with this. Microsoft you'd assume are the experts, so should be providing resolutions to these issues (and ideally testing updates thoroughly before releasing them).