Most probably it wasn't mIRC at all that got you infected. Most viral infections arrive by email these days. After their arrival it only takes two careless clicks to launch a viral outbreak on your machine. Alternatively they enter your PC because you didn't set a proper password on your administrator account.
The particular situation you refer to, with a mIRC still running you can't close, I bet you got infected with (something like) Trojan.IrcBounce, that includes a malicious copy of an old mIRC.

Trojan.IrcBounce - Is your PC infected with a mIRC version 5.7 (as the Help/About/ menu will tell you) that suddenly starts when you boot your PC? Is it hiding as TASKMNGR.EXE (not to be confused with TASKMNG.EXE or TASKMAN.EXE)? Chances are this is the virus called Trojan.IrcBounce bugging you!

From the load of reports I recieved it seems we have a firm outbreak of this trojan. Trojan.IrcBounce is the name for a collection of programs that a hacker can use to conceal intrusion and obtain administrator-level access to Microsoft Windows environments. After it is installed into your PC, it gives a remote attacker unobstructed access to your computer!

The trojan includes a copy of mIRC that hides as Taskmngr.exe actually being mIRC32.exe version 5.7. The Trojan uses this file to run all of its mIRC scripts, including Dll32.hlp, Dll32NT.hlp, Xvpll.hlp, Httpsearch.ini, and NT32.ini. Read more, and detailed removal instructions at http://securityresponse.symantec.com/avcenter/venc/data/trojan.ircbounce.html.
Note how this page tells you how the Norton virus scanner will remove the infection itself, but NOT the files in its payload. It will not remove mIRC or the registry setting starting mIRC! You'll have to do this yourself, by hand. The info page provides most info you need to do that.