Praise be to Allah.
Prayer is very important, and is the pillar of Islam that comes after the shahaadatayn (twin declaration of faith). The one who does not pray has no share of Islam, as was stated in the report from ‘Umar al-Faarooq (may Allah be pleased with him) in the presence of many of the Sahaabah. The evidence from the Qur’an and Sunnah indicates that the one who does not pray is a disbeliever and beyond the fold of Islam, whether he fails to pray because he denies that it is obligatory, or out of laziness and heedlessness.
But one who is new in Islam, who grew up in a land of disbelief and ignorance, or in a land where ignorance and corruption prevail, may be shown more lenience than others, and be excused for things for which others would not be excused.
This woman may be unaware of the status of prayer in Islam, so she should be taught and treated with compassion.
She may have some doubts, or be unaware of things, which appears to you as though she is not convinced of Islam, or is not Muslim as she should be.
What we advise you to do is give her another chance, and show compassion in teaching her, guiding her and dispelling her ignorance and doubts.
If you can take her to an Islamic centre near your home, that would be a good idea, especially if it offers educational and da‘wah activities only for women, where she can learn what she needs to know about Islam and mix with some righteous Muslim women, and learn from their good manners about how a Muslim woman should behave.
And you should be gentle with her when trying to get her to do that, as much as you can, for gentleness is not added to anything but it enhances it.
Keep trying until you think that you have given her sufficient opportunity, then evaluate the situation after that and see what her progress has been; the most important thing you should be thinking about now is her prayer and adherence to Islam.
If she does not start to pray, and does not improve her attitude towards you, then there is nothing good for you in keeping her and remaining married to her.
There is no sin on you if you leave her, if she does not adhere to Islam, even if leaving her leads to her committing suicide, or becoming lost and going astray, because she is accountable, of sound mind and responsible for herself. If she does well, then that is to her credit and if she does wrong, then that is her burden to bear. “and no bearer of burdens will bear the burden of another” [al-An‘aam 6:164].
With regard to your concern about her not bringing up children as Muslims, we advise you to draw up a legal covenant [according to the law of the land] that will oblige the wife to raise the children as Muslims and not to go against Islamic teaching in raising and educating them. This is a contract that is done in some foreign countries, and perhaps it is also available in your country.
To sum up: if your wife adheres to Islam, then keep her.
If she does not, then there is nothing good for you in continuing to live with her, so you should divorce her.
And Allah knows best.