It is logical to assume that Medicare would cover the costs since it exists to meet the health care needs of seniors. Unfortunately, whether it makes sense or not, Medicare does not pay for in-home care or a stay in a nursing home.
Is there any other government program that will fill the void?
It is not necessarily intended for this purpose, but Medicaid will pay for long-term care if you can gain eligibility. There is a low asset limit, but your home is not counted with an equity limit of over $900,000.
The fact that your home is not counted is a positive on the one hand, but you have to be aware of Medicaid estate recovery. A lien can be placed on your home if you were enrolled in the Medicaid program and you were in personal possession of the property at the time of your death.