All we know is that she's been in an asylum for a long time. We don't know exactly what happened to her, but according to Michael she has absolutely no hope of recovery.
Now that he's dead, I don't know what is going to happen to her or who's looking after her.
Since Gregson put his affairs in order before he left for Germany and left his publishing company to Edith, he most likely made Edith responsible for Lizzie's welfare. I thought we might see Edith look in on her or at least acknowledge this responsibility. Did anyone other than Matthew ever even know about her?
Once I imagined a scenario before reports came in of Michael's death where Edith had been left responsible to make choices for Lizzie and could have faced the difficult choice, if Lizzie was on some sort of life support, to take her off it which would lead to her death. Edith would be torn as Lizzie's death would mean Michael and she could be together, but at the same time he would likely not forgive her for putting the life of a woman he once loved in her own hands like that without him.
What do you think?