Instructions
Change your Paradigm to a Synergist and Saboteur (Tide Turner) at the start of the battle. Once changed, immediately use the Haste and Vigilance buffs on Sazh and Vanille. Haste will ensure your party members get in more attacks before the enemy, while Vigilance will help to prevent the "Stun" status ailment from Brynhildr's attacks.
Switch the Paradigm to a Commando and Healer (War &Peace). Auto-attack with Sazh to begin the chain bonus gauge. This will allow it to fill up when using Ravager attacks.
Switch the Paradigm to a Ravager and Ravager combination (Dual Casting) to increase greatly the chain bonus gauge. Once the chain bonus gauge fills up, switch back to the War &Peace Paradigm. This will cause Sazh to deal great damage to Brynhildr while Vanille heals the party.
Repeat Steps 3 and 4 until Brynhildr has been defeated.