Patent leather is easy care most of the time. However, it is very easy to get stains that get underneath the coating. I think that is what has happened here. I don't think you'll be able to get it clean without taking drastic measures. I've never heard of a patent leather cleaner. If it were mine, I'd try painting it. I think you can use glossy fabric paint (puff paint) on it. I can't remember. It's be a long time since I rehabbed patent. I know nail polish doesn't work!
Otherwise, you could try to remove the coating by using more nail polish remover, then attempt to clean the leather underneath, and afterward add a gloss finish to replace the coating.