It is almost always more advantageous to file Married, Jointly than Married, Separately. You do not have to live together to file jointly.
See: Is it better for a married couple to file jointly or separately?
Even if payments go directly to the nursing home, your wife will have to file a tax return - it is still her income even if the nursing home has a claim on it for her care.
Filing separately does not change your wife's income. As @ColeenD3 states above, filing separately will actually increase how much of her SS payments are taxable.
