Thanks for responding! I would be ok with 'I do not have mirrors in my house', or 'I don't have a mirror in my house', but in almost all circumstances "I do not have any mirror in my house' is not what a native speaker would do - at least in my experience.
See the discussion in http://www.bbc.co.uk/worldservice/learningenglish/grammar/learnit/learnitv114.shtml - I was looking for a more definitive reference, but that one seems good enough.
I will say, however, that the sentence "I don't have any dog in the house" makes sense - but to me it has a special connotation, of being part of a longer conversation (or argument) about a particular dog
I think Duo is looking for as direct translation as possible and counts it wrong if i'ts not exact. I don't have any mirrors in my house is correct English, but not a direct translation of the singular used here. I also was marked wrong on this one... chalk it up as singular and plural distinction