How can one deal with household members who commit abominations and persist in them, such as drinking alcohol, women not praying, not observing hijab, intermixing, and befriending Christians, while not responding to advice?

You must advise your family and explain the danger of the abominable acts they commit, and strive to supplicate for them. If they are guided, then praise be to Allah. If they are not guided, then do not grieve yourself to death over them. Avoid sitting with them while they are committing abominable acts, and if you fear that they will affect your religion, then move to another residence if possible.