Ok, you don't understand how happy I am to read this. It's like a ray of hope, as I haven't tried tackling it as an infection. Would you recommend something like savlon? or just Manuka honey?. I recently started using Listerine, but I'm not sure whether that will help.
By gunk, I assume you mean the skin on the lips?. If so, I've always had a hard time removing it all in one go. I like my lips to look even and flat without white bits hanging off it and I'll usually get very aggressive with it, resulting in very raw lips.