When does Achilles agree to begin fighting for the Achaens?

Achilles agrees to join the battle after Hector has killed Patroclus.

Achillies decided to fight at the end book 16 after he found out that Patroclus was dead and that Hector had his armor. Achillies was furious because Patroclus was his best friend. Achillies decided that Hector would pay for killing patroclus. Achillies was not exactly fighting for Agamemnon but for Patroclus.