Part of the conflict is how each sister views the other as a liability that needs to be protected because of their disability. For Annie, this is amplified by being Fia's older sister and needing to look out for her. When their parents were alive, Fia got the rather negative message that it was her job to protect her blind sister (despite being a child who was in no way equipped for this). Both are willing to do extreme, horrible things for the other. Their relationship is a complex, toxic mix of love and resentment. As hostages, they to some extent see each other as obstacles to their freedom.
Fia is suicidally depressed with impulsive, self-destructive tendencies. I enjoyed Mind Games's portrayal of her depression because it showed someone who is depressed in an angry way, and I haven't seen that very often. Many people with depression cycle through angry and depressive periods, as does Fia.
They are in fact withholding care from both sisters in order to better control them, but it goes too far to make sense. "Our spy/assassin-in-training does a murder-suicide attempt? Here are some pills and a long vacation without any mental health professionals, adults, or bodyguards! Oh, and we'll send along the (superhot) heir to the company, AND Fia's romantic rival! NOTHING COULD POSSIBLY GO WRONG."
This entire plot could have been solved if Fia had taken Hot Heir (pun intended) hostage for her sister's freedom. Which they gave her ample opportunity and motive to do. But I suppose if they had done the logical thing, there wouldn't be plot.