We've installed and configured the "Auto Respond to Message" extension. When the end user sends a message to a disconnected chat session, they immediately get an automated response indicating that the technician has disconnected, and to forward any additional requests to our help desk email address. It has been working great for us.

Yes please! I can't count the number of times end users send (important) messages in the chat window after I've disconnected from the session, and assumes I've read them. End user text entry should be disabled upon host disconnection from the session.