So if you do a split when the mite load is high, what does that do to the mite count in the hive that gets the old queen?
Now to your question, here's my answer. The best way to reduce the mite load in a hurry is to induce a brood dearth. With no brood to reproduce in the mites are soon shed from their temporary host, the worker and drone bee. The quickest and easiest way to induce a brood dearth is a walk-away split.
Sorry to hear you've been sick - I hope you're well now.
Think about the method of a split, the old queen is removed from the hive, along with a few frames of capped brood, and placed in a new hive along with new frames and/or foundation. The split with the old queen will backfill the drawn combs as the brood hatches, since the older work force is harvesting nectar faster than it can be converted into wax, reducing the room the queen has to lay eggs, this slows the reproduction rate down to a crawl for several weeks. During that time the majority of mites will fall off their temporary hosts so that as the brood production is ramped up there is a much small population of mites that can enter the brood cells and reproduce, also, the majority of thos eggs are worker brood reducing the mite reproduction because of the lack of drone comb.
In the old hive, an egg is selected and a new queen is created. Fom the period of begining the making of a queen until she hatches, mates, and begins laying is more than 30 days, again the majority of the mites have fallen from their temporary ride so there is few to begin their reproduction since the new queen starts slow the brood dearth lasts for over 60 days in the hive with the new queen, and nearly 30 days in the new hive with the old queen.
In the case of the old queen you have a guessimation of over 60% mite reduction, and in the case of the new queen, nearly 90% in mite load reduction.