Miri by Adrian Spies
Hundreds of light years from Earth, the Enterprise detects an Earth-style SOS from an unnamed planet. The first of the alternate just-like-Earth planets we'll run across. They beam a landing party down. Why is Rand there? She doesn't even have a tricorder!
They find a busted tricycle and then are attacked by a disfigured humanoid who claims it's his. He dies shortly after, like he aged centuries in a few minutes.
We meet Miri, who is scared at first because the "grups" (grownups) hurt people and eventually died off. Kirk calls her pretty, but it didn't come off as creepy to me, but more like he's flattering her in order to calm her down and get her on their side (which I myself have done with kids).
They get Miri to take them to a medical lab. The party is all starting to get lesions, except Spock. The disease seems to be a side effect of a life-extension experiment. It will take down the children too, once they reach puberty, which has been dramatically slowed. Miri and the other kids are over 300 years old.
From there, it's a race against time trying to find a cure, made tougher by the kids stealing the communicators, thus cutting McCoy and Spock off from the ship's computer.
Miri gets jealous of Janice and kidnaps her. Kirk shows her how the disease is hitting her too and she takes him to the other kids. They beat him up.He points out they're acting like grups and that their food is running out. He eventually gets them to give him back Rand and the communicators.
Meanwhile, McCoy does the Mad Scientist thing and injects himself with the vaccine they've come up with. It knocks him out cold BUT it works.
The episode has plot holes and sexism and why-does-this-planet-look-like-Earth, but... Kim Darby. She just shines (and not just from the lighting effect). She's magnetic in her every scene. She's a very understandable and even likeable character, and I put that down in large part to Darby's performance. She carried me through the episode.
I read on memory beta that Miris world is Earth from a parallel timeline that somehow ended up in the prime timeline.