Is it permissible to kill one's sister who is despaired of her reform after five years of attempts to reform her, especially with her persistence in establishing illicit relationships with young men, and her father's knowledge of that without his forbidding her?

You should continue to advise the sister who is involved in illicit relationships with gentleness and wisdom, taking advantage of influential situations. There is no harm in shunning her if shunning proves beneficial. Her father is her guardian and responsible for her, and he must prevent her from these actions; he should be advised to do so. It is not permissible in any way to kill her, because the recompense for intentionally killing a believer is Hell, the wrath of Allah, and His curse. A believer remains in the latitude of his religion as long as he does not spill forbidden blood.