Open task manager and end the mIRC process to close it if it won't let you close it normally. Then see if the internet starts working again.

Did you download mIRC from mIRC.com? If you got it somewhere else, you may have a hacked copy that is attempting to use your computer in DDOS attacks or something else that is harmful to you or others. If you are using any scripts, remove them (just delete mIRC.ini and let mIRC recreate it).