Translation:I do not have a mirror in my kitchen.
I'm with you. I would never translate this as "I have no mirror" unless I was trying to suggest the text was more than 50 years old. "I don't have a mirror" is fine, but "I have no mirrors" is equivalent to that in English.
It may be that Germans make a huge distinction between "Ich habe keine Spiegel" and "Ich habe keinen Spiegel" but most of the time, in phrases like this, German uses singular where English uses plural. "I have 10 Euros" = "Ich habe 10 Euro" etc. etc etc.
Of course even where German does use a plural, Duo often neglects to give relevant examples. I've had to track down real German speakers and ask. ;-)