The Session_End event doesn’t fire when the browser is closed, it fires when the server hasn’t gotten a request from the user in a specific time persion (by default 20 minutes). That means that if you use Session_End to remove users, they will stay in the chat for 20 minutes after they have closed the browser.
I suggest that you keep the time of the last request in the user object. That way you can determine how active the user is, and how likely it is that the user has left the chat. You can for example show any user that has not done anything for two minutes as inactive.
You can also let the chat application poll the server periodically (if you don’t do that already). This would update the last request time in the object and keep the user alive as long as the chat window is open.
You can use the onunload event in the browser to send a logout request to the server when the user leaves the page. This of course only works if the user still has net connectivity. The onunload event is also triggered when you reload the page, so you would have to keep track of why the event is triggered to use it.