Praise be to Allaah.
Islam has set out guidelines for interactions between a man and a woman who is not his mahram. It enjoins man to lower his gaze, and it forbids being alone with a woman or shaking hands with her. It also enjoins the woman to cover her entire body and forbids her to speak softly. This is what guarantees the purity of society and the soundness of the family, and it closes the door to evil and temptation. You can find the texts that speak of that in the answer to question no. 10744.
There is no doubt that your wife has overstepped these limits and has committed an action that Allah forbade when she kissed her cousin and by staying up with him when they were on their own, and by not observing hijab in front of him. Even failing to lower her gaze and avoid looking at a man who is not one of her mahrams is - in and of itself - disobedience to the command of Allah addressed to believing men and women to lower their gaze.
Taking these matters lightly on the grounds that the cousin is like a brother is a grievous mistake. How much trouble this attitude has brought to people! A woman's cousin is a non-mahram like any other stranger; indeed the harm that he may cause may be worse than that caused by others, because people take lightly the matter of interactions with him. The same applies to the relatives of the husband such as his brother and cousin. Hence the Prophet (blessings and peace of Allah be upon him) said: Beware of entering upon women. An Ansaari man said: O Messenger of Allaah, what about the in-law? He said: The in-law is death.
Narrated by al-Bukhaari (5232) and Muslim (2127). Al-Layth ibn Sad said: The in-law is the brother of the husband and similar relatives of the husband, his cousin and so on.
Al-Nawawi (may Allah have mercy on him) said in Sharh Muslim: With regard to the Prophet's words, The in-law is death, what this means is that the fear in his case is greater than in the case of others, and evil is expected on his part and the fitnah (temptation) is greater, because he is able to reach the woman and be alone with her with no one denouncing him, unlike the stranger or non-mahram. What is meant by the in-law here is the husband's relatives, except for his father/grandfather and sons/grandsons.His father/grandfather and sons/grandsons are mahrams of the wife and it is permissible for them to be alone with her; they are not described as death. Rather what is meant here is the brother, nephew, uncle, cousin and others who are not mahrams. People customarily take things lightly with regard to them, so a man will often be alone with his brother's wife. This is what is described as death and it is more important that he be stopped than a stranger, for the reasons we have mentioned above.
For more information please see the answer to question no. 13261
Even if the woman does not cover her face in front of her relatives, at least she should refrain from being alone with non-mahrams, softening her voice or shaking hands with them.
You have to explain to your wife the limits of halaal and haraam concerning this matter, and advise her and her cousin to refrain from this blameworthy leniency, because Allah will ask you about your flock which you are enjoined to guard and protect from the Fire. Allah says (interpretation of the meaning):
O you who believe! Ward off yourselves and your families against a Fire (Hell) whose fuel is men and stones, over which are (appointed) angels stern (and) severe, who disobey not, (from executing) the Commands they receive from Allaah, but do that which they are commanded
And the Prophet (blessings and peace of Allah be upon him) said: Each of you is a shepherd and each of you is responsible for his flock. The ruler of the people is a shepherd and is responsible for his flock. A man is the shepherd of his household and is responsible for his flock. A woman is the shepherd of her husband's house and children and is responsible for her flock. The slave is the shepherd of his master's wealth and is responsible for it. Each of you is a shepherd and each of you is responsible for his flock.
Narrated by al-Bukhaari (853) and Muslim (1829).
And he (blessings and peace of Allah be upon him) said: There is no person whom Allaah puts in charge of others, and when he dies he is insincere to his subjects, but Allaah will forbid Paradise to him.
Narrated by al-Bukhaari (6731) and Muslim 9143)
We hope that your wife will respond to what is enjoined upon her by Islam and will control the way she interacts with her cousin in a way that is pleasing to our Lord, and that she will not provoke her husband's jealousy, because the wise woman is the one who forgoes what is permissible in order to please her husband, so foregoing what is haraam is even more important.
May Allah help us all to do that which He loves and which pleases Him.
And Allah knows best.