All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not permissible for a Muslim woman to have a relationship with a marriageable man even if he is a Muslim, let alone if he is not a Muslim. For more benefit in this regard, please refer to Fatwa 81356.
Therefore, if it is true that your sister has a relationship with this man, then she has committed a great sin even if he did not fornicate with her. Abandoning the prayer is even a much greater sin and one of the grave major sins. Indeed you did well by advising her, and you should continue advising her. If you think that if you desert her, this will benefit her and deter her, then it is more appropriate for you to desert her.
However, if you think that deserting her will make her even more sinful, then it is better to keep ties with her and continue to advise her.
For more benefit, please refer to Fataawa 84976 and 84390.
Finally, it should be mentioned that her guardian should be reminded with his responsibility about her and that he is obliged to be firm with her and deter her from doing so. Reminding him with this matter is not backbiting at all; rather this is advice which is religiously required.