Praise be to Allah.
Strong faith must have effects which can be seen in a person's appearance and conduct. If a person persists in sin, this is a sign of weak faith.
What you have to do is try to plant the seeds of faith in her heart and help it to grow strong. What is meant by that is the kind of faith that will motivate a person to behave in accordance with the sharee’ah. Then try to instill in her a love of the hijaab and of righteous deeds, such as explaining the benefits of hijaab and how good it is, and giving her books and audio tapes, if these are available, which speak about that. One of the most important means which will help to achieve this is to put her in touch – in an indirect manner – with righteous women who wear the hijaab and try to have frequent family gatherings with righteous relatives.
If you do this, you will have tried various means of convincing her. Then you will have to oblige her in an appropriate fashion and not allow her to go out to public places without hijaab. (It is important to explain to your daughter that hijaab is obligatory and tell her about Allaah’s ruling on hijaab, even if she realizes that her mother is falling short. You have to explain it to her at a level that she can understand so that she will see that there is a difference between the rulings of sharee’ah and the way her mother is behaving. Who knows – perhaps she will advise her mother, in the moving and innocent manner of children – to wear hijaab).
With regard to the second question, there is no doubt that fathers are responsible if their wives and daughters do not wear hijaab and do not adhere to the rulings of sharee’ah, as Allaah says (interpretation of the meaning):
“O you who believe! Ward off yourselves and your families against a Fire (Hell) whose fuel is men and stones” [al-Tahreem 66:6]
And as the Prophet (peace and blessings of Allaah be upon him) said:
“Each of you is a shepherd and is responsible for his flock… the man is a shepherd over the members of his household and is responsible for his flock…”
But if a person does all that he can and strives his utmost, but does not achieve any results, Allaah will excuse him and will not punish him; on the contrary, He will reward him for his efforts, and Allaah does not cause the reward for any good deed to be lost.