It is absolutely essential that you get copies of the full veterinary record of the visit, including any diagnostic results, if you are the person that is supposed to take care of this animal. If she won’t give them then you should take the animal in to a reptile vet yourself, since you have animals of your own that could be at risk with a sick gecko around.
Honestly, in my opinion, it’s totally inexcusable that your sister expects someone who was not at the vet visit to nurse it back to health, especially because she is directly responsible for the neglect of the gecko. You need to know what the vet’s findings were and their complete instructions for care.
Vomiting in leopard geckos is a very, very concerning thing in general, but especially in this situation. You need to get in touch with the vet directly and let them know what the gecko is doing (sitting in the cold, regurgitating food, etc.), and I would say that the gecko vomiting makes this something you need to do immediately (as in as soon as they are next open).
For a sick animal, you also need to have the hot end of the enclosure around 90’F and cool end around 80. There needs to be hides on both sides of the tank and a humid hide in the middle. Make sure you are giving a multivitamin and calcium supplement on separate days. When ill, make sure you do not leave any loose insects in the tank. Mealworms are probably too hard to digest for a sick gecko because if all the chitin.