It's a nice idea, but I think a different system would work better to collect data.
The total time-limit on the conversation isn't good. A possible alternative is to limit the number of responses on each side, and have a time out on typing per-side. It's just a typing race right now and there's very little talking to the other person going on. For example, if the conversation length is 30 seconds, each side would receive 15 seconds of typing time.
"turns" of typing rather than typing at once would also give you data in the format of "statement => response" which would be much more useful as training data for some sort of AI. You'd record what was said as well as timing data (the time spent typing, how long before typing began etc), and this would allow you to mimic the user's input with your AI in a more convincing way.
Second, once a conversation ends, you can in fact just automatically start another one, and shunt the form to fill out about the previous conversation to the side.
Rather than having the conversation start when you click, have it only start on say 30-second boundaries, and you're automatically logged into a new conversation at the 30-second mark, while the form for the previous round gets shunted to the side. This would facilitate things so that humans keep chatting for longer. If the players have to click "play again" then this is a break-point at which people have to actively decide to keep playing. By assuming they're going to keep playing by default, and rolling that directly into a new chat, you'll get much higher retention.
Also, you should just let people keep playing even if no humans are available. The human in question still won't know whether they're talking to a bot vs a human, and the extra responses will be useful to train your bot. This should only matter if there's literally only one human on your site, so you should still do it.
Also other small things: a bit of CSS to make things look nicer. Horizontally center the div, because it looks much more professional than something shunted into the corner. Take the clickable options out of the dialogue box, and make them into a toolbar along the top, and add a matching footer-bar. A couple of simple things will make it look like a proper website.