Clicking on login from chat.stackoverflow.com sends me to the login page of stackoverflow.com. I log in, even though I'm already logged in, and it sends me back to chat.stackoverflow.com without being logged in. I've attempted to log out by clicking the link and then clicking the log out button but that didn't work.
I have tested the chat on Firefox 17, Safari 6.0.2 (8536.26.17) and Google Chrome 24.0.1312.36 beta.
I also visited the help page that was recommended and it offered no advice as to what might be wrong. It gave me passes on every mark.