Since (approximately) the destruction of the second temple sacrifices have ceased. Even prior to that, there was no sine qua non nature to animal blood as it relates to atonement. Biblically, only a small selection of sins could be atoned for via sacrifice, and even those could be removed through other means. We take our cue from Hosea 14:3 (some have it as verse 2) where the words of our lips (prayer, and especially the verbal recitation of the sacrificial order) substitute for the actual sacrifice.
The actual requirements for repentance go far beyond a simple sacrifice even when a sacrifice is efficacious. Regret, recompense, confession, an acceptance of blame, a decision not to repeat the action are all necessary elements to a full repentance.