(I'm a trained beauty therapist).
Extreme dry skin is quite common post surgery or after illness, can also be z side effect of medication.
Lashings of moisturiser, several times a day if necessary. Avoid all products containing SLS. (Sodium laurel sulphate). Very drying and can really irritate already dry skin.
Oils are better than creams or lotions. Many Mediterranean women swear by olive oil, both for cooking but also as a skin oil. Personally I think it smells a bit so I use Grapeseed oil. You can add a nice natural fragrance oil such as bergamot or rose. Makes it really luxurious. I use a ratio of 1 drop of fragrance oil, to 6 ml of Grapeseed.
I would also suggest a supplement, either Evening Primrose Oil or Starflower oil. Takes a couple of months for skin supplements to work.
Also, unless you've been told otherwise by your medical team, avoid low fat diets. Lots of oily fish.
Unfortunately central heating can have a very drying effect on the skin. Heating and cold winds are a double whammy.