Chlorine bleach will kill e cuniculi spores, in case the head tilt your friend’s bun got was caused by e cuniculi. A 10% solution of bleach seems to be the best thing, e cuniculi spores are highly resistant in the environment. If the cage is wooden, some of the bleach will be absorbed by the wood and you’ll need to air it out very thoroughly before you place a bun in it. Bleach will also of course kill bacteria. The normally recommended disinfectant for use around rabbits, Virkon S, is ineffective against e cuniculi spores but kills lots of other pathogens.
If it’s really cold outside it wouldn’t hurt to put the cage outside for 72 hours or so before you bring it inside, but it not a method that takes care of all possible pathogens, so thorough cleaning will (as you state yourself) still be required. If the cage is dirty it needs to be cleaned first to remove the dirt so the bleach will be able to reach all parts of the cage. Chalky residue from pee is easiest removed with vinegar and water.