It doesn't mean 180 of those people want to join the chat. Just look at this site, as for now (when I'm writing this), 562 users online, 0 in chat. Something went really wrong with IP.Chat and i guess IPS officials are fully aware that something needs to be done. It's been like this months after months, great user activity, no chat activity. It's a huge fail as I see it, but it doesn't need to stay like this. Back to your case, being site owner seeing a full chat, just encourage him/her to buy larger chat deals. It means greater income for IPS. I can see the issue, 20 people clicking (or hit-and-run) chat, everyone is inactive, preventing active users from joining. I mean, an inactivity limit could stop this right? No one wants to join an empty chat room, and inactive users will most likely "drag" active users to the chat. Also as pointed out earlier, I don't think people do much "hit-and-run" by purpose, as it is now they cannot stay on chat and browse forum at the same time without having to open a new tab or window. If these people could browse forum while also writing on chat, chat would thrive more. I doubt we would have an issue with inactive users.
Lets take this scenario: A user is writing a topic, a large post (takes him 10 minutes to write), while writing the topic, he is also active in chat and awaits a reply there. He gets a reply and goes straight over to chat box to respond back. After this he goes back to finishing up his thread, still following the ongoing conversation in chat - all happening on the same page, no extra tabs or windows. That's where we should be in 2012.
So when you say put the chat window on every page, you don't mean load the chat room itself off the bat. You mean show an area for the chat with a "click to join" button?