What should a husband do with his wife who refuses to cover her hands and feet with socks and gloves after applying henna, especially in her brother's house or when traveling where there is much mixing and her hands are exposed to men?

A wife must cover her feet, without any difference of opinion. As for her hands, even though there is a difference of opinion regarding them, she must cover them if there is a fear of fitna (temptation/strife), especially if they have naqsh (designs/engravings) and adornment, due to the prevalent corruption among people in this era. Similarly, Islamic law dictates the obligation of covering the face and hands in this corrupt era, to prevent slander and preserve honor.

Therefore, your wife's uncovering of her hands or feet in front of non-mahram men is a reprehensible act (munkar) that you must change with gentleness and leniency. If she does not respond, then prevent her from going out until she adheres to the Islamic guidelines in her dress and speech. If you fear a definite harm, then you may abandon the inkar (enjoining good and forbidding evil), but you must alert her that the husband's brothers are non-mahram to her.