... that I registered with, which is 'floccinaucini'. ...
The way your post reads, it sounds like you have already registered the nickname of "floccinaucini"
with Mibbit's NickServ.
If so, good.
Then the answer to your question is:
Type: /nick floccinaucini
You can type that in any IRC window/chat and the irc server will try to change your nick to "floccinaucini". The leading slash - "/" - makes it a command.
If you have already registered that nick, then the next step will be to IDENTIFY with NickServ. It is quite possible that you will have to do this within 30 seconds to 1 minute, else NickServ will force change a new "guest" type nick on you. The idea is to 'protect' the registered nick.
To identify with NickServ, type: /msg nickserv identify your_password_here
substituting in, with whatever password you registered with the nick.
(To get a reminder of how to do this, you can type: /msg nickserv help identify
and it will give you brief help, right in irc. }
Now back to your original post .....
When you said, "that I registered with" - if you really meant to refer to the username that you registered
here on the forum, then everything above does not apply.
Reason: registration on this forum is not connected in any way with registration of nicknames on an irc server.
It is easy to assume that they are, because first you come here and identify yourself with the forum, then perhaps you are using a link that you find here.... you click it, and get a web interface to the chat. Right?
(The reasons and history behind this would digress too far from your original question....)
So... if you have NOT registered your nick with NickServ on Mibbit, then follow patio's direction, to register your nick there.
Or, you can get brief help, when online and connected to mibbit's irc chat server, by typing: /msg nickserv help register
It will respond with instructions.
Here is a link to the instructions that patio mentioned:
http://wiki.mibbit.com/index.php/NICK