How can a woman obtain a divorce from her husband who does not adhere to the teachings of Islam, does not provide for his family, and refuses to divorce her except on the condition that she relinquish their daughter, and is it true that she cannot divorce him except with his consent?

It is not permissible for your sister to remain with her husband who does not pray, because he is a disbeliever. If he did not pray at the time of the marriage contract, then the contract is invalid. If he stopped praying after the contract and continued to abstain until her waiting period ('iddah) expired, then the marriage is annulled. She is not legally required to obtain a divorce from him, but she should seek to obtain a divorce in official documents. The custody of the child belongs to her mother. The husband should be advised to repent and pray. If he insists [on not praying], then she should refer the matter to the court to request a divorce due to non-maintenance and harm.