Zakat is payable on gold and silver jewelry, even if it is not for business purposes, provided it reaches the quantity of nisab and has been possessed for one lunar year or more.
Yes, the zakat on your wife's jewelry is payable by her. If she does not have enough money to pay the zakat, then either she could sell some of the jewelry, or you could give her enough money to enable her to pay the zakat.
It is reported that a woman came to the Prophet (may Allah bless him and his Household and grant them peace) with a daughter of hers. On the daughter's arm were two thick bracelets of gold. He asked her, "Do you give the zakat of this?" She said, "No." He said, "Would it please you for Allah to put them around you as bracelets of fire on the Day of Resurrection?" Thereupon, she removed them and threw them before the Prophet (may Allah bless him and his Household and grant them peace), saying, "They are for [the cause of] Allah and His messenger."
[Narrated by Abu Dawud, Tirmidhi, Nasa'i, Ahmad and others.
Hafiz Ibn Hajar said, in Bulugh al-Maram, "Its isnad is strong."]
A'ishah (may Allah be pleased with her), the wife of the Prophet (may Allah bless him and his Household and grant them peace), says, "The Messenger of Allah (may Allah bless him and his Household and grant them peace) entered upon me and saw bands ( i.e. rings without any stones) of silver on [the fingers of] my hand. He said, "What is this, O A'ishah?!" I replied, "I put them on to make myself beautiful for you, O Messenger of Allah!" He said, "Do you pay their zakat?" I said, "No." He said, "They are sufficient for you [as a share] of the Fire [of Hell]."
[Narrated by Abu Dawud]