A man remains in custody on suspicion of murder. Police confirmed he is related to Ms King and was also a resident at the Kent home.
Murder probe detectives said a revolver has been recovered from the scene.
Emergency services were called to the home just after 9am yesterday following reports that a woman had been seriously injured, police said. Officers and ambulance crews attended but Ms King had died.
Detective Chief Inspector Simon Werrett, who is leading the investigation, said: “At this stage we are not looking for anyone else in connection with the investigation and our inquiries are ongoing.”
A scene guard has been put in place at the premises while forensic searches are conducted.
Douglas Carswell, the Ukip MP for the area, said he was “very shocked” by the news, adding that Walton-on-the-Naze was usually a peaceful place.
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He said: “It’s very shocking. If you have got a loved one in a care home, you expect them to be safe. People need to know the exact circumstances of what happened, it is extremely disturbing.
“Walton-on-the-Naze is supposed to be one of the most peaceful and tranquil places there is – elderly people want to be in a peaceful and tranquil environment.
“Of all the places you expect this to happen, this is the last.”
Nearby resident John Knights said security was tight at the care home.
Retired financial adviser Mr Knights, whose wife Marlene runs the Bucket and Spade guest house a short distance from De La Mer House, said: “My mother had to go into De La Mer about two years ago and I can tell you that it was quite tricky to get in and out of there.
“Security was tight. You couldn’t just go up to the building and get in.
“You needed a security code to get in. It had electronic key pads on the door, so to get inside you needed to know those.
“Whoever was responsible must have known the code.”
A spokeswoman for the care home said there would be no comment until staff have spoken with police.