His wife does not cover her hands – should he threaten to divorce her?

We have explained in the answer to questions no.
11774 and
21536 the ruling on
women covering their faces and hands, and that it is obligatory for them to
cover them in front of non-mahram men. 

Allaah has enjoined woman to be obedient to her husband, and
has made men qawwaamoon (protectors and maintainers) of women. They are
supposed to direct and look after women as a leader directs and looks after
his people, because of the physical and mental characteristics with which
Allaah has distinguished them, as well as the financial duties that He has
enjoined upon them. Allaah says (interpretation of the meaning): 

“Men are the protectors
and maintainers of women, because Allaah has made one of them to excel the
other, and because they spend (to support them) from their means”

[al-Nisa’ 4:34]

Ibn Katheer said:  

‘Ali ibn Abi Talhah said, narrating from Ibn ‘Abbaas: “Men
are the protectors and maintainers of women” means: they are in charge
of them and the woman should obey her husband in the matters in which Allaah
has enjoined obedience. Obeying him means treating his family well and
looking after his wealth. 

Tafseer Ibn Katheer, 1/492. 

If the wife goes against
his wishes and refuses to obey him, the husband has to take a gradual
approach in handling the matter. First he should admonish her and remind her
of the punishment for disobeying his command. If that does not succeed, then
he moves on to forsaking her in bed. If that does not work, then he may hit
her, but in a manner that is not severe, and there is nothing wrong with
also threatening her with divorce. 

Allaah says (interpretation of the meaning): 

“As to those women on whose part you see ill-conduct,
admonish them (first), (next) refuse to share their beds, (and last) beat
them (lightly, if it is useful)”

[al-Nisa 4:34]

Shaykh ‘Abd al-Rahmaan al-Sa’di said: 

“those women on whose part you see ill-conduct” means,
those who refuse to obey their husbands and disobey them in word or deed;
they should be disciplined gradually. “Admonish them (first)” means,
explain the ruling of Allaah concerning obedience to one’s husband,
encouraging obedience and warning against disobedience. If the wife stops,
then that is what is wanted. Otherwise the husband should forsake her in her
bed, by not sleeping with her or having intercourse with her, until the
desired result is achieved. Otherwise he may hit her, in a manner that is
not severe. If one of these methods achieves the desired result and the wife
obeys, then “seek not against them means (of annoyance)” i.e., you
have got what you wanted, so do not keep rebuking them for things that are
in the past, or draw attention to faults that will cause trouble if you
mention them.  

Tafseer al-Sa’di, p. 142. 

You can seek help in admonishing her and advising her by
means of useful tapes and books which speak of the obligation of hijab for
the Muslim woman, and explain the seriousness of disobedience and its
consequences in this world and in the Hereafter. 

You have to be gentle and kind with her. The Messenger of
Allaah (peace and blessings of Allaah be upon him) said: “Allah is kind
and He loves kindness and confers uponkindness that
which he does not confer upon harshness anddoes not
confer upon anything else besides it (kindness).”  Narrated by Muslim,
2593.   

And he (peace and blessings of Allaah be upon him) said:
“There is never any kindness in a thing but it adorns it, and kindness is
never lacking from a thing but that makes it ugly.” Narrated by Muslim,
2594.  

And Allaah knows best.