Whilst I agree with the general consensus that Dettol is not a great anti-mite treatment and wouldn't use it for that purpose myself, there are a couple of points which I think are worth a comment in relation to the question.
Dettol - traditional liquid in a bottle - active ingredient is chloroxylenol (C8H9ClO). It can be toxic to animals, even humans, if ingested or inhaled and is banned as a pesticide in Australia where it was used to control cane toads. Dettol brand other substances such as disposable wipes product contain other chemicals which may be toxic so beware confusing brand names with active ingredients.
The fact that a substance is not licensed for a use does not automatically make it illegal to use it. Guidelines are not law, but generally for laypeople they should be followed. Professionals (vets, doctors etc) make judgments on use and can prescribe 'off licence'.
There are many substances which are generally considered safe (often labelled GCS) which have been used for a very long time and have never had modern testing but are accepted whilst there are other (safe) substances which are not licensed or marketed for a use because it is just not worth the manufacturer spending the large amounts of money on obtaining approval.
Bleach (thin/old-type sodium hypochlorite) is not a detergent but often detergent and odour producers are added to make it more effective/nicer. It has the advantage that the final breakdown product is salt, so is quite good for cleansing where it cannot be guaranteed to have been rinsed away.