I had assumed that she had once, previous to the shows start, that she had had a meaningful affair with Doakes, that after the fact transcended into respect and friendship.
(I assume a lot.)
LaGuertta doesn't have to blackmail Deb. Deb is her subordinate. She can make Debra's life a living hell if she chooses to.
LaGuertta's motives are pure, if a little zealous.
Some asshole shat on her friends name with lies and she wants to clean it up.
(Remember when she tried to collect money for his funeral and everyone laughed at her?)