I believe I would start with tea tree oil on the dog. Demodectic mange can be hard to clear up, but the obvious first step is to rid the dog of the mites. This passage from a dog owner that I saved some time ago in a Google search seems like a reasonable approach:
"... bathe the animal thoroughly with mild soap and warm water. Clip away any excess hair or shave the area. Combine one teaspoon of tea tree oil in a cup of water and put mixture into a spray bottle shaking with each use. Spray the animal down thoroughly and allow them to stand for up to ten minutes. Use a paper towel to dab off the excess moisture, then dispose. Saturate a cotton ball with tea tree oil and dab onto the stubborn areas. This treatment should be repeated daily for at least one week or until signs of the mange are gone. Keep the animal isolated during treatment and disinfect the kennel or pet carrier as well."
I believe Neem extract would work also, and I would think that LiceBGone and/or FleaNtick Bgone also should work. Orange oil might be too harsh for broken skin spots, so I think I'd try the tea tree, Neem, or the Bgone enzyme products first before the citrus. Because it's a puppy, I'd watch for signs of irritation or adverse reactions with anything you try. That may be even more of a reason to use the mildest possible effective treatment.
As for the environs, I probably would wash what softs I could in hot water, spray the hard areas with an orange oil or d-Limonene solution, such as diluted Orange TKO. I can't say from memory whether beneficial nematodes can help with this type of mites in the yard/lawn or whether predatory mites can control the demodectic mites in the outdoor environment. A little Web research should help answer that question, or mabye someone on the forum knows that answer immediately. Finally and obviously, take a look at the dog's diet and evaluate whether a change might be in order; you probably already did that. Good luck.
In theory, theory and practice are the same; in practice, they aren't -- lament of the synthetic lifestyle.