What is the way to mend the relationship with a sister who refuses reconciliation and insists on a personal apology, despite repeated attempts to communicate with her and apologize?

You have done well in your commitment to upholding kinship ties, and you will be rewarded for it. It has been confirmed that whoever maintains ties with his relatives while they cut him off is as if he is feeding them hot ashes, and he will continue to have a supporter from Allah against them. The one who truly maintains ties is not he who reciprocates, but rather he who connects with his relatives when they cut him off. Maintaining kinship ties is a religious obligation and an act of worship to Allah. We advise you to fear Allah, maintain ties with your kin, consider your sister's circumstances, and show her mercy. This requires fulfilling her wish for you to go to her and apologize. So be the better of the two disputants who initiates the greeting, for the Prophet, peace and blessings be upon him, said: "It is not lawful for a Muslim to shun his brother for more than three days... and the better of the two is the one who initiates the greeting."