The AKC isn't trying very hard. There are plenty of documented cases where they have re-registered puppy mills after brief suspensions, despite the mills still being in trouble with the law on animal cruelty and other charges. They have also registered mills at the same address as ones they recently de-registered, when a relative living at the same address submits a new application. If the AKC were serious about protecting dogs and breeds, they would strictly limit the number of papers they sell each year, to a number reflecting readily available information about the numbers of real homes available (i.e. no sudden geometric increase in papers for Dalmatian puppies just because the mills are eager to cash in on the movie). If they took this approach, they could make certifying breeders a very competitive process, and the papers would really be worth something. But they're not willing to take the revenue cut from stopping the flow of phony papers.
They have also registered mills at the same address as ones they recently de-registered, when a relative living at the same address submits a new application. In essence, you're right, although AKC doesn't register mills, they register litters, dogs, and kennel names. But once a person is suspended, it is easy to transfer his/her dogs to a relative, and this does happen.
Those of us who are ethical breeders don't like the puppy mills either, but AKC is never going to move towards closing them down. Why would any business cut off one of it's income sources?
AKC is not in the business of protecting dogs; it is in the business of registering them. It is up to the people who care to close down the mills.