This is perhaps off topic but I wanted to post it for others to find when they have the same problem I did.
Problem: you can’t get chat to open in Gmail.
Even if the icon for it is showing, clicking it does nothing.
The icon, usually in the lower left of the page:
Google the problem and the best you will come up with is this page, which was no help to me.
Solution: close and reopen gmail. Then click the icon. BTW that can fix a lot of problems in gmail or calendar. I find both can responding after a few days of being open in Chrome for OS X.