Will my brother die killed for having killed a woman with his car, with the blood money paid to her family, and does the Hadith: "Whoever kills will be killed, even if after a while" apply to him?

If the killing was intentional, then your brother bears the sin, and he is liable for retribution (qisas) unless the heirs of the deceased pardon him. He shall be killed in the same manner he killed that woman, according to the Almighty's saying: "And if you punish [an enemy], punish with an equivalent of that with which you were harmed." However, if the killing was accidental, then there is no sin upon him, according to the Almighty's saying: "And there is no blame upon you for that in which you have erred." In this case, the blood money (diyah) of the deceased is incumbent upon the killer's 'aqilah (male relatives on the father's side), and he must emancipate a believing slave or fast for two consecutive months, according to the Almighty's saying: "And never is it for a believer to kill a believer except by mistake. And whoever kills a believer by mistake – then due is the freeing of a believing slave and a compensation payment presented to his family [of the deceased] unless they give [it up as] charity."