@RubySlipppers is one with the Body, and correct. It is the scene where Kirk talks his computer to death in
Return Of The Archons. In the
Hall of Audiences:
Clue 1: A cry for help is not how this scene starts.
Computer Landru at the start of the scene is confident that it will prevail. The scene ends with it pleading for help.
Clue 2: Nobody is killing everybody is the theme.
Computer Landru is discovered to be not a man but a machine, so, it has "no body". It says, "Obliteration is necessary" about our heroes. Clue 5 has more on the Landru and its killing theme of the scene.
Clue 3: Kirk sounds a little annoyed after fixing the problem.
Both Marplon and Reger were cowards in the end which did not endure them to Kirk. Last line is from Kirk, "Well, Marplon, you're on your own now. I hope you're up to it. (to Lawgivers) And you can get rid of those robes. If I were you, I'd start look for another job."
Clue 4: Spock is to be treated as an infection (for the first time).
The line given in Clue 2 continues to Kirk and Spock with "The infection is strong. For the good of the Body, you must die." Later in the series, Spock (and everyone else) is trapped inside the Space Ameba and its trying to kill them like an infection.
Clue 5: Kill you all is the solution.
Computer Landru states several times during the scene that it will kill Kirk, Spock and anyone else that has knowledge of them. It says, "All who saw you, all who know of your presence here must be excised. The memory of the Body will be cleansed." Etc.
Clue 6: Kirk defines a different Prime Directive, even though he ignores the Starfleet Prime Directive.
The Starfleet Prime Directive is first established in this episode in a prior scene.

Kirk and Spock in conversation:
KIRK: Landru must die.
SPOCK: Captain, our
Prime Directive of non-interference.
KIRK: That refers to a living, growing culture. Do you think this one is?
Why is Kirk even bothering with this planet and its people? Anyway, computer Landru states "The good of the Body is the Prime Directive." Then Kirk confuses computer Landru with his own redefinition of its prime directive, "Then the evil must be destroyed. That is the Prime Directive. And you are the evil." Then it fries its program.
Clue 7: The scene opens with Kirk loudly announcing who they are (which is technically not correct.)
As given in your above post, at the start of the scene in the Hall of Audiences, Kirk shouts that they are the Archons which is technically not correct since they are not from the Starship "Archon" rather they are from the Starship "Enterprise" so they are Enterprisians or something like that.
Clue 8: One thinks the patient is healthy. One thinks the patient is dying. Neither are doctors.
Computer Landru declares "The body is. It exists. It is healthy." Then Kirk declares "The body is dying. You are destroying it." Neither computer Landru nor Kirk are medical doctors (at least it was never established that they are.)
Clue 9: Clue two has a sub-clue done twice.
A play on words. "Body" is seen twice in Clue 2: No
body is killing every
body is the theme. "Body." Get it?
Clue 10: Clarifying Clue One, a cry for help is how this scene ends.
Computer Landru's last line is a cry for help to original Landru who programmed it, "Help me! Help me! Help me! Help me!" (Fizz, bang, boom.)