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 is His slave and Messenger.
First of all, we thank and praise Allaah Who guided you to Islam and we ask Him to make you firm and steadfast on His Religion until the Day you meet Him.
We advise you to seek beneficial knowledge, perform good deeds and befriend righteous and pious women, as these are the best things that help a Muslim woman remain steadfast on her religion.
As regards your daughter, if she has not yet reached the age of puberty, then she is a Muslim, in which case, the child follows the parent who is best in religion as we clarified in Fatwa 81957, so she is ruled to be a Muslim following you in religion.
However, if she has reached the age of puberty, then she is ruled to be following the religion that she is upon, so if she chooses to be non-Muslim, then she is a non-Muslim [a disbeliever], and if she chooses Islam, then she is a Muslim.
For more benefit on the signs of puberty, please refer to Fatwa 83431.
Therefore, we advise you to supplicate Allaah as much as possible to enable her to follow the truth and observe the Hijab. Besides, you are obliged to teach her the matters pertaining to her religion (Islam) and be soft and gentle with her so that she would follow the truth. It is more appropriate to follow with her the manner of story telling and the like. In discussing with her, you should stress the good traits of Islam and the benefits of abiding by the Hijab. In this matter, if you can seek the help of some pious girls at her age, then this is good.
If you do your duty in calling her to Islam but she chooses to be non-Muslim [chooses a religion other than Islam], then you are not sinful, Allaah willing.
In case you marry, your daughter becomes a stepdaughter and a Mahram to your husband.
Finally, it should be noted that there are disadvantages for Muslims residing in non-Muslim countries, especially the children, so we advise you to try to migrate to a Muslim country to reside there, as this is one of the best things which will help you nurture your daughter on righteousness.
For more benefit, please refer to Fatwa 86405.
Allaah Knows best.