I'll be honest, I understand none of this. Taking it line by line here.
"You are a 15 year old girl who, along with other people, were given a second chance of life."
OK, presumably you're saying that the protagonist and their allies in this story all died and got a second chance at life, or at least the protagonist and Hatsu. No idea how.
"You live with a strange illness that worses in snow and can kill in blizzards. It also appears randomly."
No idea what this means. Is the illness from being reborn? What does it mean in gets worse in snowy weather? Like if you're in the snow, or even if you're inside and its snowing, or... Anyhow, just move somewhere it doesn't snow. Also, when you say it appears randomly, does that mean people get it randomly, or sometimes it's symptoms don't appear in snow and you're fine, or what?
"You live with one of these people, Hatsu. You have a jeweled heart, known as your Angel Of Darkness Heart. If it shatters/breaks, you die. Cracking is a time of emotion."
Like most things, the destruction of the heart does mean death. The name is weird, and I don't know what it means. Also, does that mean every time of emotion permanently fucks with your heart?
"You have to wear a necklace that has the infinity sign on it. If it is removed from your neck, you die."
Uh... OK, whatever.
"You are attacked by people who chose a different path of the second life, the Shadows."
Nothing wrong with that, just so many questions raised.
"You have to survive."
A common goal.
No idea what you mean by ideas. Change the name of the heart and title, I guess. Nothing more unless you explain a lot more.