Which is better: the niqab or the khimar, for a wife who wishes to wear the niqab outwardly, while she does not know the fundamentals of her religion and limits herself to performing the obligatory duties?

One must prioritize correcting their creed. Your wife's interest in wearing the niqab is good; teach her the fundamentals of religion thereafter. The niqab covers the face, and the khimar covers the head; both are required for women. Covering the head is obligatory by scholarly consensus when non-mahram men are present. As for covering the face, there is a difference of opinion regarding its obligation if uncovering it does not lead to fitna (temptation/strife). Scholars have adduced evidence for the obligation of covering both the head and face from His Almighty's saying: "And let them draw their headcovers over their bosoms" [An-Nur:31], where the khimar covers the face and the bosom when lowered. And His Almighty's saying: "They should draw over themselves some of their outer garments" [Al-Ahzab: 59]. Also, His Almighty's saying: "And as for elderly women past childbearing who do not expect marriage, there is no blame upon them if they discard their outer garments in a non-displaying manner of adornment; but to modestly refrain is better for them" [An-Nur:60]. The permission for elderly women to discard their outer garments indicates the obligation of hijab and face-covering for young women, and modest refraining is better for them.