God the Almighty says in the chapter of Al-Nisaa,
“You are forbidden to take as wives your mothers, daughters, sisters, paternal and maternal aunts, the daughters of brothers and daughters of sisters, your milk-mothers and milk-sisters, your wives’ mothers, the stepdaughters in your care– those born of women with whom you have consummated marriage, if you have not consummated the marriage, then you will not be blamed– wives of your begotten sons, two sisters simultaneously– with the exception of what is past: God is most forgiving and merciful” (Quran 4:23).
This noble verse illustrates the categories of women that are prohibited for a man to marry because of relations established by blood, nursing, or marriage. According to the verse, some women are permanently prohibited for marriage while others are temporarily so.
The inquirer in this case is classified in light of the above-mentioned verse as a permanent mahram (prohibited for marriage) to the inquirer’s late daughter through the marital bond. The Prophet (peace and blessings be upon him) said, “If a man conducts a marriage contract with a woman, her mother becomes permanently unlawful for him to marry” i.e. by virtue of the marriage contract.
The majority of scholars and jurists from the four schools of jurisprudence have agreed that after conducting a marriage contract, the wife’s mother becomes permanently prohibited for her son-in-law, whether or not the marriage was consummated.
Based on this and in reference to the question, it is permissible for the inquirer to live in the same house with her son-in-law. In this case, he is considered a mahram who is allowed to take care of his elderly mother-in-law and attend to her needs. She is allowed to do without hijab in his presence, travel with him and behave as she would with her own son.
God the Almighty knows best.