I've just read the series and while it is very depressing I think it's executed quite well.
Kaori probably admired Shizuku because the latter was so energetic and lively, which for a sickly girl like her was probably a sort of ideal or foil to herself. From that angle it makes sense why she seems to have invested her last summer on earth to rescue that same girl from her depressed state and return her back to the vigorously living girl she once was. In a way, this reminds me of Useless Princesses, where one character is basically living through the other, although quite literally in this case. Her initial plan of just fading away from Shizuku after summer then kinda makes sense, her objective is to leave Shizuku as revitalised as possible without adding additional trauma. Of course, believing that she could just disappear without causing trauma is naive, but I think it's at least believable motivation.
It also explains why she is willing to use underhanded methods in order to force Shizuku out of her shell and force the meeting with Ruri as well, Shizuku is basically the avatar through whom she wants to continue living, so freeing her from her emotional baggage is Kaori's number one priority, which she pursues relentlessly. She might even be willing to risk Ruri for that, though I'd say that she also believed the meeting would help Ruri, and considering that she still seems to nurse that trauma and isn't really able to forget, that might even be true.
I'd imagine Ichihara and Kaori were in the same class, which is why they know each other. Ruri and Kaori probably spent a lot of time together in the hospital and became friends that way. That kind of raises the question why Kaori claimed to have just heard about the bullying through rumours, because that seems unlikely considering how close she is to both other girls involved. It looks like there's still a piece of information about their elementary school days that she hasn't shared yet, which is interesting.
Overall I would be really surprised if Kaori doesn't die in the end, the entire story seems to be guided by her trying to fix things before her death and Shizuku overcoming trauma, so her death will likely be the final piece of trauma that Shizuku will need to overcome before riding off into the sunset.