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. We ask Allaah to exalt his mention as well as that of his family and all his companions.
First of all, we ask Allaah to greatly reward you for your good intention in guiding this woman. In principle, a Muslim should only marry a pious woman. The Prophet said: "A woman is married for four qualities: her wealth, her honourable lineage, her beauty and her religion; so choose the religious one." [Al-Bukhari and Muslim] Therefore, if one is tolerant in this matter, this could lead to many problems and unpleasant consequences. So, if your wife is as you mentioned in the question, that she has a relationship with another man, and that she stole some money from work, in addition to her not listening to your advice and guidance, then she is sinful; she disobeyed her Lord and neglected the rights of her husband. As regards her claim that she repented and regretted what she did, if you are pretty sure that she has sincerely repented and that is performing good deeds, then it is more appropriate to keep her as a wife, especially if you are blessed with children with her. However, if you are sure that she has not sincerely repented and her matters are doubtful, then divorce her; because a woman who has a relationship with a man other than her husband, it is not worth being the wife of a chaste Muslim husband, and it is not safe that she will not give birth to children who are not from her husband.
By the same token, we do not know what you mean by saying you started the procedures of divorce, if you had uttered that you divorced your wife, or that you wrote that you divorced her intending to initiate divorce, then she is divorced. Taking her back or not depends on, as we have previously mentioned, whether she has sincerely repented or not. However, if you did not utter divorcing her or that you did not write the word divorce while intending to initiate divorce, then she is still considered as your wife.
Allaah Knows best.