Just like some people, some chicks are hatched with birth defects. That might be an injury but it looks like a birth defect. I agree with Azygous, it probably won't make it. It sounds like it is not eating or drinking on its own. It might be a mercy to end it's life now. That is a hard decision to make.
If you want to try, I'd use a medicine dropper or your syringe and put a drop of liquid on the tip of its beak. If you try to force food down its throat you will probably strangle it. It should swallow a drop of liquid on the tip of its beak if it has any chance to survive. Keep giving it liquid until it stops swallowintg on its own.
What liquid to use? I use hummingbird liquid, it is full of high energy sugar. Some people use sugar water. You can usually get electrolytes at the feed store, they may go under different names. The idea is to get liquid in it so it doesn't dehydrate and also get some energy in to so it feels like eating and drinking. If it cannot soon eat and drink on its own it will not make it. That hummingbird liquid has saved a grown hen and a couple of chicks, but by the time you get to that stage they often don't recover.
Hate that this happened to you.