From the little sister's reaction, Kaori has been trying to erase her existence from her family the same way Shiuku has from hers. Thinking that that she's being kind, Kaori's been trying to put distance between herself and everyone she loves in the hope that people won't be so sad when dies. She even hid the fact that she was dying from Shizuku and wanted to pretend that she was merely going away after playing lovers for a time.
In Chapter 19, Shizuku realized how blinded she was to how many people around her loved her and worried for her and would care if she disappeared. In Chapter 20, we realize that Kaori is similarly (perhaps willfully) blind and still doing the same things that Shizuku used to.
Of course her little sister reacts like she does in this chapter. Her beloved older sister is dying and for some reason doesn't want to spend any time together. She's jealous of the one person her older sister hasn't abandoned.
Lay off the imouto, people!