If goats are like sheep, then it will eventually be passed as the young suckle, or she treads on the trailing end. Never be tempted to pull it out yourself. Our rule is that if we haven't got it 24 hours later then we give a long acting antibiotic to cover her for internal infection and repeat after 3 days. There is also the option of something like Synulox which is put inside the uterus, but usually if there is residual infection after the afterbirth is out.
If there is a foul smell then consult your vet straight away.
I'm sure some goat folk will be along with goatier experience.