It would be "Das ist nicht eine Küche." Because "nicht" contradicts "is." If you said "Ich habe keine Küche." It would mean "I have no kitchen." which contradicts "kitchen" as you said. "Ich habe nicht eine Küche" means "I don't have a kitchen" which is a far more common way of saying it in English.
"Die Küche" is feminine indeed and "eine" is used correctly.
I think you were confused by the "Das" which means "this" or "that" and it is not the neutral article in this case.
Das ist eine Küche. = This/That is a kitchen.
Das ist die Küche. = This/That is the kitchen.
In this exercise, Küche's article is "eine" and it is positioned right before the noun.
Make a hissing sound, like a cat, for the "ch." Your tongue should press against the roof of your mouth while you blow air over it. It's also common to hear the "ch" sound pronounced like a hard "k" (like in my last name) in some parts of Germany. You can also hear it pronounced like "sh" in other parts. It depends on the dialect.
The female voice sounds like the German sentence is "Das ist deine Küche". Since that also makes a proper German sentence and distinguishing between 'ist eine' and 'ist deine' is pretty hard, I think they should accept both.
There have been similar comments about this from FIVE years ago.
I think 'Diese' would only work if directly in front of the noun. You would have to use 'das'. Das ist eine Küche.