Although you viewed your sister as active and in good health before COVID, being 76 with LHS, will complicate recovery. Many, much younger, are struggling months on end. But, this doesn't mean she can't recover.
It's important for her to see a physician who has been working with LHS patients. Perhaps a well informed Geriatric specialist, who understands senior needs, and I wouldn't rule out other specialties.
General health has to be monitored closely, including dietary needs, and supplements. A 40lb weight loss is serious. Her physician should be very alarmed. How can she heal if she's nutrient challenged? Losing that much weight quickly, at her age, will affect general health, muscle and bones. This will add to weakness and fatigue.
She's not healing she's declining. If her doc doesn't have any answers, he should have referred her somewhere else. I hate that seniors aren't always given the attention they need. We're not disposable.
Some patients are given stimulants for brain fog and fatigue, with noted improvement, but this would be tricky for someone 76. But, there may be safe alternatives.
Her entire body has taken a serious hit. I'd make sure all levels, e.g., ViD, B12, Iron, etc., are within recommended range, along with Thyroid levels. Cardiac and lung changes post COVID need f/u monitoring. Nothing should be missed, or dismissed.
The older we get, the longer it takes to recover from serious illness. Even seniors who receive anesthesia for minor surgery, often find they have to deal with brain fog and sometimes persistent cognitive changes. As healthy as we may seem, we are fragile and things can turn on a dime.
It's good you're being an advocate for your sister. As bad as she feels, while also fighting depression, she might not be able to fight for herself, or make the best choices.