I get almost sick to my stomach when I see her letters
It can be difficult, can't it? Personally, I would discard the letters without reading them. Our MIL is in a nursing home and whenever there is a health issue, the nursing home telephones us. Perhaps, you could mention this to the director in charge of the home to make you aware of any "change" in your mother or any "task/directive" that requires your attention. Most nursing homes are excellent in this mode of care.
Then, weekly I would write chatty letters - of course you would not be addressing her "concerns" because you would not know them. Maybe all she needs at this time in her life is contact, albeit through the written word.