I'm so sorry to about Gracie's nail sticking out at an abnormal angle and her limping. Since the nail is sitting at an abnormal angle it is probably partially avulsed from the nailbed and a partially avulsed nail is painful.
The nail does need to be removed.
Every time she walks on the paw the moving nail piece injects bacteria and dirt into the nail bed area. That will lead to infection. I understand that she may not let you touch it, I'm sure it hurts too much. She will need to see a veterinarian for quick sedation and removal of that nail piece.
In the meantime the less walking she does on that foot the better.
You will need to keep her indoors, if she goes outdoors it may get wet or dirty which would definitely lead to an infection.
You also need to keep her from licking. If she licks it will likely become infected.
So if she's trying to lick then place an elizabethan collar.
Do not use peroxide to clean the area. It is very irritating and can actually damage healthy tissue. If she will let you you can soak her paw in epsom salts or a dilute solution of warm water and antibacterial hand soap to remove any dirt, then dry well and put a very small dab of antibacterial ointment on the nailbed if she will let you. Sometimes if the nail is very loose it may even fall off when soaked.
If her toe or nailbed area becomes swollen or tender then she may need oral antibiotics from her veterinarian. They can determine that when they remove the nail piece.
As her nail grows out you may need to carefully trim and reshape it so she doesn't catch it again and damage the nailbed.
Please let me know if you have any further questions.