Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I am sorry to hear that Milly is getting very thin even with a normal appetite, is lethargic, and has a dry cracked nose.
If she is sick enough to lose weight then it would make sense that she is lethargic.
I suspect that she has been sick longer than you realize and she has had subtle changes in condition that were hard to realize because you see her every day, but once the symptoms became more obvious then you noticed.
When we see weight loss with an apparently normal appetite we need to worry about disease processes that either don't allow her to digest and/or absorb the food she is taking (such as diabetes, liver disease, primary intestinal or pancreatic disease including cancers) or those that lead to a higher metabolic rate such as cancer.
If she is having trouble taking in appropriate nutrition then her body will prioritize the parts of her body to support and her skin (including her nose) and coat are much less important than her brain and internal organs, so it makes sense that her nose isn't normal.
In an older dog the more common diseases we see include liver and kidney disease, primary intestinal disease or if she were drinking more and urinating a tremendous amount, diabetes. Sometimes early on with diabetes we only see weight loss.
Both diabetes and primary organ failure are often treatable or at least manageable.
Diabetes is controlled with diet and insulin therapy.
Organ failure can be managed with diet changes, supplements, fluids, and medications.
It sounds like some testing needs to be done, including a complete blood count and biochemistry profile as well as a urinalysis.
If those are not diagnostic then it is time to move on to more specialized testing. I recommend checking vitamin B levels (cobamalin and folate) to look for signs of intestinal disease, a test for pancreatic insufficiency called a TLI, and an abdominal ultrasound to look for signs of changes in the walls of the intestines as well as evaluating her organs. She may need an endoscopy to collect biopsies of her intestines if they look abnormal.
In the meantime you might try feeding her a higher calorie prescription food called Hills a/d or Iams Maximum Calorie from your veterinarian and make sure she has access to plenty of fresh, clean water.
Please let me know if you have any further questions.