Sorry to hear of your mite attack! I have only had one orchid get mites and I believe it had them when I bought the plant (at least hidden in the media). It did kill that orchid but did not spread any further. I did find the spider mite infection and moved it away from the rest of the collection.
I think that repotting is a good idea. Then you will know that there is no eggs to hatch. But be sure to closely inspect the roots for eggs. You want to be sure to not allow them to move to the new media.
Good luck! I find the phal equestris is tough, like you said!