Keep working on it.
Mites do need a live host to survive but the eggs can continue to hatch without effective treatment. Keep the soaks going and try the reptile spray and run him through a towel with the spray on. Soak again afterwards and repeat the process.
The areas around the eyes, nose and mouth are the tough spots to work on.
Fortunately, I've never had to deal with them so my advice isn't anything more than prevention type advice with some my own thoughts on how to attack the problem.
I feel for you as a Bredli is one of the easiest keeps when things are going well.