IIRC Jeanette's explanation about why she sued and why she supposedly assumed Kate's life were separate in that scene?
Jeanette sued because Kate basically ruined Jeanette's life by outing her as someone who saw her in the house but didn't go to the cops about it. In the back and forth shouting over this, Kate then admitted that she was angry about her life being stolen while she was locked up in Martin's basement. Jeanette then elaborates on that saying, as you wrote, that she wanted to experience being in the 'sunlight'.
This reasoning was only about why she assumed Kate's life and not why she sued her. Naturally she did lose all the perks of being "New Kate" once actual Kate returned, and this probably did make Jeanette angry, but, for all outward appearances, the law suit was based on the character defamation brought about by Kate's interview.