I would think that one could fill a spray bottle with water and Nix and spray it into the cracks of the wodden cage. That might work.
You might want to find a temporary home for your dragons and isolate the wooden enclosure from them and any other herps for a while. Treat the wood and enclosure with something (Nix, Black Knight etc). I'm no expert but I've been told that mites leave the host animal to lay eggs. In the cracks of the cage or in the substrate would be good spots for that. They can take a while to hatch (a few monthe I've been told). If you keep the cage (& the mite eggs) away from any food source (like your dragons) and treat it every so often with anti-mite stuff...you should be able to kill off the mites.
If there is anything the nonconformist hates worse than a conformist, it's another nonconformist who doesn't conform to the prevailing standard of nonconformity.