My wife and I performed Hajj this year. We slaughtered only one sheep as hady (sacrifice for Hajj). Having returned home, I asked a Shaykh about this, and he said that we are still liable to slaughter another sheep for my wife. I delegated a concerned institution in Mecca to slaughter a hady on behalf of my wife. Taking into consideration that I am keen to make our worship perfect, is our Hajj valid?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

Your Hajj is valid, by the permission of Allah. Regarding the hady, there should be a differentiation between the obligatory and the recommended one.

For a pilgrim who is performing Ifraad Hajj and who has neither violated one of the restrictions of ihram (ritual consecration) nor abandoned one of the mandatory acts of Hajj, offering hady is only recommended.  It was authentically reported that the Prophet, sallallaahu ‘alayhi wa sallam, offered one hundred camels as hady. [Al-Bukhari and Muslim]

An-Nawawi said:

Scholars unanimously agree that it is recommended for a person who heads towards Mecca for Hajj or ‘Umrah to offer a hady of livestock, slaughter it there, and distribute its meat among the needy who live within the Sacred Precincts of Mecca. It is also recommended that he offers a good, sound and valuable one. [Al-Majmoo’]

The majority of scholars are of the opinion that a six or seven month-old sheep is enough for hady.

On the other hand, hady is obligatory when there is a reason for it, such as making a vow, performing Tamattu' or Qiraan Hajj, compensation for hunting or violating the restrictions of ihram due to an ailment in the head (necessitating shaving), or abandoning one of the mandatory ritual of Hajj.

So, if both of you did something necessitating hady and offered only one sheep, then this does not absolve any of you of liability. That more than one person share in offering a single animal for obligatory hady is invalid according to the opinion of the Maliki and Shafi‘i scholars . Ibn Al-Atheer narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said: "A sheep or camel shall be slaughtered for only one person." [Jaami’ Al-Usool]

Thus, each one of you is required to offer a hady, with all relevant conditions fulfilled.

It is permissible to authorize an institution or someone to slaughter the hady on your behalf in Mecca.

Allah Knows best.