When does the atonement fasting for the questioner's brother, who caused a car accident that led to the death of a child, and who began fasting on 7/9/2019, conclude?

If the accident occurred without negligence on your brother's part regarding speed or safety precautions, then there is no guarantee (of recompense) or expiation upon him. However, if the accident was due to negligence without intent to kill, two things are required of him: expiation (freeing a believing slave or fasting for two consecutive lunar months) and blood money (Diyya) for the victim's heirs, which is borne by the 'Aqila (the male relatives on the father's side).

The lunar months are what are considered for calculating the two months of expiation, due to the Almighty's saying: "They ask you, [O Muhammad], about the new moons. Say, 'They are measurements of time for the people and for Hajj.'" So, your brother must know the start date of his fasting according to the lunar months. If he began his fasting on the tenth of Muharram, for example, he should complete fasting the entire month of Safar by the sighting of the crescent, and then fast ten days of Rabi' al-Awwal to complete thirty days for Muharram. However, if he began his fasting from the beginning of the month of Muharram, then fasting Muharram and Safar will suffice him, whether they are full or incomplete.