This may be old news for some, but when monitoring chat at the Information Help Desk be sure you are set as “Staffing” both the text and chat. Otherwise, you may be signed in but not actually monitoring anything (which causes the chat to disappear from the website). When signing in, Connect with both areas checked:
When signed in, if you see a blue alert icon next to “I’m Staffing”, click on the “I’m Staffing” button and select both options: