That actually may not be Scaly Face, as in the after-bath photo her beak doesn't look effected. It's impossible for anyone to diagnose her through photos, and that's why it's so difficult to give you advice, because if she has Scaly Face or some other type of mite infestation they won't go away and will only get worse without proper treatment; if it's not a mite infestation and you treat her for mites, it's not going to help whatever the real issue is...So it's tough. Mites are definitely likely though, based on the conditions and circumstances in which you found the bird...
*****One very important note that you need to take very, very seriously with such a small, sick Budgie...If the medication that you have coming to you is Ivermectin, which is the appropriate treatment for most mite infestations in birds, you absolutely must be EXACT about the dose that you give the bird, whether given orally, by injection, or topically on the bird's skin, doesn't matter, it must be EXACT! Ivermectin is an extremely strong, dangerous, and toxic poison, and it has killed many a pet due to the too high a dose (I've seen birds, reptiles, and rodents killed by too much Ivermectin, it's nasty stuff, but unfortunately it's necessary and the best we have currently to treat certain parasites)...So you are probably going to have to find a scale of some kind to weigh the Budgie, as the Ivermectin dosage is based on weight, as are most anti-parasitic medications. And with such a small bird, it only takes a single milligram too much of Ivermectin or other antiparasitic medications to kill the bird. So just be extremely careful, and be extremely exact in your dosing...