All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The Global Arabic Encyclopedia says about this disorder: “Schizophrenia is a serious mental illness marked by severely impaired thinking which cannot be predicted … Schizophrenia is one of the most common mental disorders … and many schizophrenics are delusional and act as if they live in an imaginary world. They may hear sounds that others do not hear and may believe that these sounds carry messages from important people ... They show inappropriate emotions, such as laughing in sad circumstances. Some schizophrenics avoid their families and friends, and basically they talk to themselves or to the sounds which they hear whereas other people do not hear them.” [Summarized]
Whether the schizophrenic person is labelled as insane or not, his wealth is still liable for Zakah. Hence, if your wife has not reached the degree of insanity, then she should pay the due Zakah herself or delegate someone to do it on her behalf. You stated that she does not understand anything about Zakah. If you mean that she does not know the religious rulings pertaining to Zakah, then her ignorance of these rulings does not exempt her from the obligation of paying the Zakah. It is still incumbent on her to consult scholars and seek their help in calculating the Zakah for her; Allaah, The Exalted, says (what means): {… so ask the people of the message if you do not know.} [Quran 21:7]
On the other hand, if her condition has reached insanity, she is still not exempted from paying the Zakah; the insane person’s wealth is liable for Zakah according to the opinion of most Muslim scholars. Al-Mawsoo‘ah Al-Fiqhiyyah Al-Kuwaytiyyah (Kuwaiti Encyclopedia of Jurisprudence) reads: “Muslim jurists unanimously agreed that illness does not constitute a valid cause to exempt the person from paying the Zakah because health is not among the due conditions for the validity of Zakah payment in general. However, they held different opinions regarding insanity and whether the insane person is exempted from the payment of Zakah (during the period of his insanity) or not, given that insanity is a form of illness. Rather, it is one of the most serious illnesses, as highlighted by Ibn ‘Aabideen. The majority of Muslim jurists maintained that liability of Zakah remains attached to the insane person and he is not exempted from paying Zakah, given that all the due three conditions for the validity of Zakah are met: freedom, Islam, and having full possession of the property.
This opinion has been reported on the authority of ‘Umar, ‘Ali, Ibn ‘Umar, ‘Aa’ishah, Al-Hasan ibn ‘Ali, and Jaabir . It was also adopted by Jaabir ibn Zayd, Ibn Seereen, ‘Ataa’, Mujaahid, Rabee‘ah, Is-haaq, Abu ‘Ubayd, Abu Thawr, and other scholars.
In this case, the appointed legal guardian over the insane person's wealth is obliged to pay the Zakah and other financial liabilities such as fines and obligatory expenses on behalf of the insane person because this is among the financial affairs that are managed by the legal guardian.”
Finally, the husband is not the one addressed with the obligation to pay the Zakah on his wife’s wealth. We could not find any Muslim scholar who held that the husband is made legal guardian over his wife’s wealth when she is legally declared mentally incompetent and denied the right to dispose of her property if this is done for her own benefit (as in case of the young, insane and the foolish regarding the money related matters). Hence, the legal guardianship over her wealth is assigned to her father or grandfather or whoever is appointed by the Sharee‘ah court, if available, or the Islamic Center in non-Muslim countries because in this case the Islamic Center serves as substitute for the Sharee‘ah court.
Allaah Knows best.