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I'm sorry to hear about your dog. It's fairly rare for a dog to have ear mites, but they do often have bacterial or yeast infections in their ears.
I would definitely clean the ears by using an ear cleaner specifically made for dogs, available from any pet supply store or online catalog or vet. Make sure the canal is filled, then massage the base of the ear well, then allow your dog to shake the excess debris out. Wipe the exterior gunk off. Wash the ears daily, but gently, since it is so swollen.
You are eventually going to need an antibiotic or anti-fungal from the vet, depending on what the culprit is causing this. The vet will put a sample under the microscope to determine this. The day of your vet exam, don't clean the ears that morning, so some debris will be built up for a good sample.
Until then, continue cleaning daily and do your best to discourage scratching and further irritating the ears. I've included a link here (scroll down) to see how to properly clean the ears in case needed:
Do not continue to use neosporin in the ear. It will not help this, nor is it formulated for safe use in dog ears.
Best of luck. Let me know if I can help further.