She went to Darent Valley Hospital in Dartford before she was admitted to the Hospital for Tropical Diseases.
The trust said the UK was rabies-free and staff would be unlikely to consider rabies unless a patient reported wild animal contact in an at-risk country.
Doctors responded to information given by the patient at the time, it added.
The woman, who has not been identified and is still receiving treatment, had been bitten by a dog in south Asia.
In a statement, the Kent NHS trust said some hospital staff members were being vaccinated against the potentially fatal disease.
It said: "Although there are no cases of rabies being passed by human-to-human contact, the five members of staff that came into close contact with the patient are being vaccinated as a precautionary measure."
The trust also said an investigation had begun into the circumstances around the patient's attendance at the emergency department.
It said it was also working closely with the Health Protection Agency (HPA).
Rabies is usually transferred through saliva from the bite of an infected animal, with dogs being the most common transmitter of rabies to humans.
More than 55,000 people are thought to die from rabies each year, with most cases occurring in south and south-east Asia.
Professor David Brown, a rabies expert at the HPA, said only four cases of human rabies acquired from dogs, all from abroad, had been identified in the UK since 2000.
A woman with rabies twice visited an accident and emergency unit in Kent before she was diagnosed and admitted to a London hospital.