How should one deal with disagreements amongst sisters in Islam? How should one deal with the situation when the sister doesn't want any further contact with me, despite the fact that this behaviour is un-Islamic?

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
All the Muslims are obliged to live with love and intimacy avoiding any kind of dispute and disagreement. The religion of Islam urges Muslims to cooperate with each other and to exchange sincere advice to one another. All these good ends could not be achieved if any lasting disagreement or hatred existed within the Muslim community. Therefore, every Muslim is obliged to spare no efforts to settle all kinds of disputes with his Muslim brother and try his best to live in hormony with other Muslims, particularly if he was mistaken. If any other Muslim is mistaken or if he is responsible for any disagreement among Muslims, in this case such person should be admonished and should be informed of the bad consequences of causing dispute within Muslim community, using all the good means, soft words etc. If he accepts the call to settle the dispute then this is what is intended otherwise his right as a Muslim still remains upon other Muslims. Therefore, Muslims should maintain good relations with such a person, particularly if he is a blood relative or a neighbor. It is narrated in Sahih Muslim from Abu Hurairah that a man said to the Messenger of Allah: "I have relatives with whom I try to keep the ties of relationship but they severed relations with me; and whom I treat kindly but they treat me badly; I am gentle with them but they are rough to me." He (Blessing and peace be upon him) replied, "If you are as you say, it is as if you are feeding them hot ashes, and you will be with a supporter against them from Allah as long as you continue to do so" [Muslim].
Abdullah Ibn Amr reports that the Prophet said: "Al-Wasil is not the one who recompenses the good done to him by his relatives, but Al-Wasil is the one who keeps good relations with those relatives who had severed the bond of kinship with him." [Reported by Imam Bukhari].
Allah knows best.