OK, sounds lke she is hallucinating. The sister is not being very skillful in dealing with this, as though WE can't see what she sees, her experience is exactly what it would be if it were really happening --- so if one isn't tactful in how one tries to re-assure her, it can be avan more frightening for her. Ideally, she needs to be seen by a Psychiatrist, maybe a good GP if one atends the home.
Many old people function very well so long as their health is fine, but can be easily tilted into confusion and hallucinations by many different factors --- maybe an infection, like a bladder or chest infection and a fever ( maybe that's why she feels so hot and feels sure the fan isnt working properly ) --- or as a side-effect of many different medications she may be taking for other reasons. SO what she needs is first a good physical checkup to be sure there isn't any physical condition needing to be treated before it gets worse. Then there are several anti-psychotic sedative medications she can be given which will both help her to feel more peaceful, and help to clear up these hallucinations. If it is related to a dementia --- and this condition may appear so gradually, espeially wih someone living a quiet life in a home, that it may already be quite advanced before it is noticed --- if it is in the earliest stages there are some medications which may slow its advance, but otherwise one would use the same anti-psychotic tranquillizers as before mentioned.