It is not essential for one to declare one’s Islam before witnesses in order for it to be valid

I am a young man who is married to a Muslim woman but her family is not Muslim. She became Muslim and learned some soorahs and how to pray via the internet. After we got married I asked her: at whose hand did you become Muslim? She said: I became Muslim by myself. I said to her: You have to utter the Shahaadahatayn, so she uttered them before me. Is that correct?
 Please note that we got married in the mosque before the Shaykh and a number of witnesses, and she got a certificate proving that she is Muslim.

Praise be to Allaah.

Firstly: 

In order for a person to become Muslim, it is not essential for him to declare his Islam before anyone. Islam is a matter that is between a person and his Lord, may He be blessed and exalted. If he asks people to bear witness to his Islam so that it may be documented among his personal documents, there is nothing wrong with that, but it should be done without making that a condition of his Islam being valid. 

This has been explained in more than one answer. Please see the answers to questions no. 13698, 11936, 655, 6542 and 6703 

Secondly: 

No marriage contract is valid without a wali for the wife, but no kaafir can act as a wali for a Muslim woman. If she has no Muslim wali then the qaadi or the imam of the mosque or the local mufti may act as her wali. 

In the answer to questions no. 7714 and 7989 you will find more details on the ruling on women in kaafir countries who do not have Muslim walis. 

We ask Allaah to bless you both and to join you together in goodness.