Scratching dachshund

Are you quite, quite sure the problem is not dog fleas?

I have known vets miss this obvious cause of itching.

Changing the dog's diet has had no effect, so I would start looking at what her bed, or rug is made of.

Check the ingredients in the foodstuffs used, dogs can be allergic to milk, including milk powder and to wheat. Most animal food contain colouring agents too - another well-known source of allergies.
Animals, like humans, can be allergic to synthetic materials, especially those like polyester that contain formaldehyde.

Animals can also be allergic to the washing-powder their blankets are washing with.

Is the dog a town animal with restricted access to the outdoor world and exercise?

I ask because I have known indoor cats that became so frustrated that they scratched themselves incessantly, causing rashes and scabs that no vet could do anything about, as the cats in question were owned by a person who refused to believe that a cat could be bored and need exercise, if not outdoors, then strenuous exercise indoors.