If you have a malware infection that is not cleared by a factory reset that means it's managed to insert itself into the /system partition. So to fix that you need to reflash the phone with a clean copy of the official firmware. I don't know much about LG so don't know where you find this or the tools - perhaps LG or your carrier might be able to help, otherwise hopefully someone here will be able to advise.
Android malware actually takes the form of trojans, i.e. malware hidden inside some other app, rather than viruses. I'm not saying this because I'm a pedant, but because it's important: you will have been infected by something that you installed yourself that had malware hidden inside that, so you need to make sure you don't just reinstall the same thing after you have cleared it up.
BTW I'm taking your word that you've correctly diagnosed the problem, even though none of your security software can identify it. But we do often get people who see something they don't understand (including a network connection) and jump to conclusions, so if you have any doubts do post more information about the symptoms if you want people to give an opinion.
