What is the sufficient and encouraging answer for my veiled wife - to whom I will present the answer - so that she may be convinced to wear the niqab, as she is hesitant about it?

If the headcovering (khimar) that the wife rejects is that which covers the head and neck, then it is not correct to describe her as "veiled" (muhajjabah). The hijab that Allah commanded believing women to wear includes the khimar, as stated in His saying, "And let them draw their headcovers (khumurihinna) over their bosoms (juyubihinna)." The khimar is a religious obligation because a woman's hair is considered awrah (private parts that must be covered) by the consensus of Muslims. The verse was revealed because women used to let their headcovers hang down their backs, leaving their chests and necks uncovered. So they were commanded to cover their bosoms.

The sister must understand that the hijab in its entirety, including the khimar, is a command from Allah and His Messenger. A Muslim woman has no choice but to submit to it outwardly and inwardly, and to believe that what Allah has commanded is truth, justice, and good. Allah denied faith to those who do not submit to what the Messenger brought, outwardly and inwardly. The complete hijab, including the headcovering (khimar), is among what the Messenger brought from his Lord. Therefore, it is not permissible for a believing woman to be displeased with it or to feel uncomfortable about it. Allah has made the hijab for the benefit of women, to preserve them and protect their honor, as He Almighty said: "O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused." The meaning of "That is more suitable that they will be known" is that they will be known as chaste and honorable women, so they will not be subjected to harm or harassment.