Once you've treated the cage with provent a mite, it should kill all of the eggs and mites. You may want to treat again in like a month? to be sure, but it should keep you from getting repeaters.
I'd also suggest you treat the floor around the cage as well as mites can travel extreme distances.
I think black knight II works the same way, but I've never used it.
And if you have pet arachnids, move them out to a different place...provent a mite will kill arachnids.