No. No date was ever put on Kaien's death, so we do not know if it correlates to Ichigo's birth. Second, Kaien was fused with the Hollow Metastacia, which was reformed in Hueco Mundo after its defeat. There, it was eaten by Aaroniero, the ninth Espada, who absorbed them. Aaroniero was still alive years after Ichigo was born, so no, Ichigo can not be the reincarnation of Kaien.