Is it morally permissible to divorce in order to qualify for Medicaid assistance to pay for nursing home costs? My wife has dementia and will soon need nursing home care. Her savings would be gone in 9 months, and then they would come for my assets.

You describe a difficult situation faced by many today. A brief column such as this cannot explore all the moral issues involved here, but the bottom-line answer is, no, you should not divorce. 
 
The well-known axiom that the ends do not justify the means applies here. And while the “end” of trying to save your money, presumably to give it to your children, is a good and understandable end or goal, one cannot sin in order to obtain it. 
 
What is the sin involved in what you ponder? Fundamentally it is either to divorce, which God hates (see Malachi 2:16) or it is to lie. 
 
Regarding divorce, it is essential to recall the vow you made which is very pertinent in exactly the case described here. The vow said, “I take you to be my wife, to have and to hold, from this day forward, for better, for worse, for richer, for poorer, in sickness and in health, until death do us part.”: Clearly sickness and poverty were anticipated as a possible scenario in the vows you both made. 
 
But one might argue, “We are not really getting divorced, it is just a legal move regarding civil marriage. We will still consider each other as spouses.” But in this case, a lie is being told to the State for the purpose of Medicaid funds. 
 
Either way, it seems that what is proposed is that one do evil (sin) that good may come of it. This is not a valid moral solution to an admittedly difficult and painful issue. 
 
In recent years, Long Term Care Insurance has been a solution to some of this, but for an older person, this new device is seldom much help since, if they have it at all, the premiums were high and the payoff low. 
 
I pray it might be of some consolation to recall that the goal in life is not to die with a lot of money in the bank. The goal is to die in holiness. God has promised the Kingdom to those who are persecuted for the sake of righteousness, and who have done what is right, even at high personal cost.