First you pray. The issue of Church authority is the greatest obstacle to conversion (It certainly was for me!). It takes great humility and courage to admit that Jesus Christ established a visible Church on earth. Just keep praying for your wife and loving her in the Lord. And always keep in mind the efficacy of God’s grace. Your loving testimony to the truth will communicate itself to her eventually, and then her heart will begin to open to the working of God’s Holy Spirit and his truth.
To help your wife with the issue of Church authority, perhaps you can use your shared beliefs as the basis for your discussion. For example, you can agree with your wife that the Church is the people of God. Talk about how this church or this body is arranged. Tell her it has bishops and presbyters, people in authority that we are to obey (Heb. 13:17). Tell her that the Church is a spiritual reality, but one that is manifested visibly. Reassure her that it is not “the Church or a relationship with Christ”, it is both. Jesus never intended them to be apart.
Explain to her about how the “Church in this world is the sacrament of salvation, the sign and the instrument of the communion of God and men. It’s both visible and spiritual, a hierarchical society and the Mystical Body of Christ (1 Tim 3:15, Luke 10:16, Matt 18:17, John 21:15-17, 1 Cor. 14:33). She is one, yet formed of two components, human and divine. That is her mystery, which only faith can accept. (*Catechism of the Catholic Church * no. 779-780). I will be praying for you both.
Here are some books you might suggest she read:
*Evangelical is not Enough * by Thomas Howard
By What Authority? by Mark Shea
*When Only One Converts * by Lynn Nordhagen (Editor)
*Search and Rescue * by Patrick Madrid (how to bring your friends and relatives back to the Church)