Fullcircle,
Interesting question, brought a lot of thoughts to me.
Something stood out to me. You say her partner locked her out of the apartment because she was tired of the illness. This does not sound complete.
One does not react to someone's ill-health by locking them out.
She may have broken up with her and asked her to move out?
NYC is an expensive place to live. If she insists on staying there for her career but is in a shelter, sounds like her priorities are not quite straight.
Re your other daughter, at 40, this whole thing did not start yesterday. Sometimes the sibs can see what the parents don't. I saw my sister smoking and climbing out the window to meet her boyfriend. They have no idea to this day.
It is said, that "the most powerful person in a house, is the sickest person in it".
Maybe she is sick of playing second fiddle to your eldest.
Having said all that, I am sorry that you are hurting.