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.
If your friend had left her marital home without a sound reason, then she is Naashiz (a recalcitrant wife) and she loses her right upon her husband to spend on her during the period of her recalcitrance. For more benefit, please refer to Fatwa 90507.
Besides, as long as a woman is still a man's wife she does not go out of the bond of marriage with him just because he does not spend on her or because he does not have sexual intercourse with her regardless whether the period is long or short. Rather, she does not go out of the bond of marriage with him unless he divorces her or that an Islamic judge divorces her from him as we clarified in Fatwa 148027.
Since this case is in the hands of the judge, then he is in a more suitable position to study the matter and take the appropriate ruling.
As regards your question as to how it is possible for the husband to provide his wife with her needs if she does not go back to her marital home despite his continuous attempts to convince her to do so, then if by ‘the needs’ you mean spending on her and clothing her and the like, then the answer is that all these matters are not an obligation upon the husband until the wife gives up her recalcitrance.
Finally, it should be noted that a Muslim is in principle innocent, so it is not permissible to accuse this woman that she has an illicit relationship with another man without evidence; Allaah says (what means): {O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin.}[Quran 49:12]
But if it is confirmed that she has a relationship with a non-Mahram man, then it becomes an obligation to advise her to repent and admonish her to give up this relationship. Of course, the sin is greater if a woman has such a relationship while she is married.
Allaah Knows best.