Welcome to health care Magic
Your nail plate is split due to dog bite. Your nail matrix from which new nail is formed is near your nail fold. According to your description that area is alright. Normal nail will grow out and it usually takes 3-6 months.
I hope the dog is immunized for rabies, otherwise you need to take anti rabies vaccine
. If you have not taken tetanus
shot in last 5 years i will recommend a TT injection as well. The bruise
you are having what we call subungual hematoma
will disappear in 1-2 weeks time.
Hope this helped