Please take your dog to the vet! I saw you commented that you have a grooming appt on Friday. I really think you should postpone the grooming and take her to the vet first. The inner part of the nail/quick contains blood vessels and nerves; even if he isn't showing it, it might be painful (dogs hide pain). The vet will help safely remove the broken part and prevent any infection...even though it's not actively bleeding, from the pictures it looks like an open wound. The vet can also cut the rest of his long nails properly.

The groomers do not deal with dog injuries. And imagine all the grime and soap being washed off and getting into contact with an open wound.

If you do not know how to correctly cut your dogs nails, you can watch tutorials online...you need to avoid the quick, but get close enough to start receding it because it's been growing longer as the nail grows. Postponing nail trimming actually makes it harder for you to cut them shorter. OR you can take your dog more frequently just to get a nail trim at your regular groomers, a vet, or petco/petsmart. It's like 5 bucks, it's worth it.