Do your worst, you freak!
Like all Eidolons, Brynhildr inflicts Doom on Sazh at the beginning of the battle, and instead of defeating her, Sazh must fully fill up her Gestalt bar.
Sazh should apply Vigilance to both himself and Vanille to stave off Knockback, and Haste to quicken their actions. SYN/SAB combo builds the Gestalt meter well. The party can heal with a paradigm like COM/MED and then begin building chain with two Ravagers.
It is simpler (and often easier) to have Sazh exclusively as a Synergist, and use Vanille as a Medic and Ravager. To begin, Sazh should cast Haste and Vigilance on himself, and then use the same on Vanille. After that, he should cast Faith, Bravery, and whatever En-Spells he may have. The player might have him switch to Ravager now and then, but constantly use the Synergist abilities and switch to SYN/MED whenever HP is low, or if the buffs expire.
An alternative to quickly fill the Gestalt bar is to start the battle with SYN/SAB and cast the respective buffs and debuffs, mainly Haste and Vigilance for both characters, Faith for Vanille and Enfrost with Bravery for Sazh. Then the player should now move between COM/RAV to go full offensive, and the bar will fill easily, and COM/MED to keep overall offensive while Vanille quickly heals any damage with her Faith buff and then switch back to COM/RAV to finish filling the bar.
Another alternative is for Sazh to use Fortisol before he meets up with Vanille, since it will cast Bravery, Faith, and Haste right from the start. The player should have Dualcasting, which will have Sazh and Vanille as Ravagers, as their default paradigm, then switch to War & Peace to restore health.
Brynhildr is a Shieldmaiden and a Valkyrie in Norse mythology. In the Völsunga saga, she was sent by Odin to decide a fight between two kings, but was punished by the god after helping his least-favored one then banished her to the top of mount Hindarfjall in the Alps. She must sleep within a ring of fire until any man rescues and marries her. Sigurd entered the castle and awoke Brynhildr by removing her helmet and cutting off her chainmail armor.
The two fell in love and Sigurd proposed to her with the magic ring Andvaranaut. She appears as a main character in the Völsunga saga and some Eddic poems treating the same events. Under the name Brünnhilde, she appears in the Nibelungenlied and therefore also in Richard Wagner's opera cycle Der Ring des Nibelungen.