I agree with Fairorchids, there's no need to cut the roots. Cutting the roots opens up the living tissue for infection, which you really want to avoid. Plants are very resilient and can take care of themselves for the most part.
The wiry part of the root is the stele, which may still be functioning to draw water into the plant if not too badly damaged. So DO leave those wiry root parts attached! I honestly don't see the harm in leaving the outer mushy layer either, as it acts to wick water into the plant when attached. I'm not sure how much this function changes after the root dies though. Look at this link on the topic for a better explanation from Ray with a diagram:
Firm Brown Roots
The only reason I would cut any portion of the roots off (living or dead) is if they had a nasty fungus growing on them and it was spreading. In that case, I'd leave the plant out of the media to allow the wound to completely heal before repotting. This would be a severe case and you would probably want to also use some sort of anti-fungal treatment isolated to the infected area. This is what I would do based off my horticultural knowledge and even then, I would ask an experienced orchid grower who has dealt with this before.
Anyways! Your plant looks healthy to me and the roots look fine. Good planting!