What is the technique from Karin that made Tsunade recover that fast? Even the side effect of her technique subsides after the bite?
That's one of Karin's powers - biting her lets people absorb her chakra and its healing properties.
From Karin, in Narutopedia:
she not only has incredible healing capabilities, but can also heal others by letting them bite her and absorb her chakra. The potency of her healing chakra is great enough to even restore a bisected Tsunade to full health.