Brackett's pips suggest a vice admiralrank, which contradicts the rank given to her in dialogue. This could, however, refer to a position as opposed to a rank - she could have been responsible for a fleet of starships.
The character was not given a first name in the episode, but the novelization gives her name as Ruah Brackett. In the novelization, Brackett appears in a scene after Picard meets Sarek and gives him permission to go to Romulus. The novelization also mentions that Brackett was at Starfleet Academy with Picard.