The problem with that is that dish soap is very harsh and full of toxic chemicals. Not something you would want to put on your lips (or your hands). The combination of borax and hydrogen peroxide has specific properties. Also, not all moisturizers are the same - pure lanolin HPA is very different from the products other people have used (on another forum people did use Lansinoh and they got good results).
I read practically every post for the past 4 years, and no one has tried the *specific* method and combination of products that I'm using. I chose those products very carefully due to their respective properties.