Embedding the Mibbit client into your own website is simple.
For more details, follow the guide on the wiki
Here is the code for a simple example. Just replace the channel parameter.
There are more options you can include in the URL (See wiki), but this should get you started.
<iframe width=500 height=280 scrolling=no border=0 src="http://embed.mibbit.com/?channel=%23trivia"></iframe>