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 is His slave and Messenger.
If the Islamic marriage contract of a woman is conducted with her husband, then she becomes his wife. One of the most important Islamic purposes of marriage is the durability of the marriage and the achievement of the desired benefits from the marriage. Whenever it is possible for the married life to be stable, then this is better, and one should undertake the reasons that would help in achieving that.
Indeed, you have done well by endeavoring to achieve this, may Allah reward you.
The Sharee’ah has given the woman the right to ask for Khul'u [divorce in return for compensation] if she hates her husband and fears that this will prevent her from fulfilling his right upon her.
This is what the wife of Thabit Ibn Qays did and her story is reported in Saheeh Al-Bukhari.
If the case is as you mentioned that your wife asked for Khul'u simply because she does not like your mother, then she is wrong; but she is not obligated to live with your mother; rather, she has the right to be in a separate accommodation in any case.
For more benefit that a wife is entitled to have a separate accommodation and that she is not obliged to live with her in-laws, please refer to Fatawa 253297, 88654, 384583, 297622, 253297, 189065 and 137042.
You are not obliged to accept her request for Khul'u. We think that you should continue to try to convince her and convince her parents that the marriage bond continues. You should frequently supplicate Allah earnestly, as He ordered us to supplicate Him and He promised to answer our supplication; Allah Says (what means): {And your Lord says, "Call upon Me; I will respond to you." Indeed, those who disdain My worship will enter Hell [rendered] contemptible.} [Quran 40:60]
There is nothing wrong in appointing some righteous people again to mediate as Allah may make them a reason for reconciliation. If that is possible, then praise be to Allah. Otherwise, it may be better for you to accept her Khul'u and look for another woman to marry as Allah may compensate you with a better wife, and do not grieve about what happened. Allah Says (what means): {...But perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah Knows, while you know not.} [Quran 2:216]
For more benefit about Khul'u, please refer to Fatawa 89039, 297739, 131953, 276691, 205168 and 168545.
Allah Knows best.