Actually Asami makes sense.
After she goes through what she does with her father at the end of season 1 she has no family left and she could and probably does view Korra and Mako as her family and would hold tight regardless of Mako being a douche.
She would still have anger and react to it emotionally though. She got a double whammy of being betrayed by her only girl friend, as well as her boyfriend cheating on her. Because she had no family, or anyone. It would make sense for her to feel torn. To not want to lose contact, because they are all she has. But also feel so much anger and sadness.
The writers chose to push Asami to the back, and she pretty much didn't react at all to the events that happened. The second Mako was single, she was ready to get down. And showed she wanted to be in a relationship with him again. The plot never gets resolved, and just off screen between seasons, we are supposed to accept she lost those deep feelings for Mako.
Asami overall was poorly handled. Why is why the transition from Korra and her being best friends, and then later lovers also felt clunky.