Cool cover of chimney on the house, not attached (5-4)
I believe the answer is:
fancy-free
'not attached' is the definition.
(fancy-free means not romantically attached)
'cool cover of chimney on the house' is the wordplay.
'cool' becomes 'fan' (fan can mean to cool down).
'cover of' means to remove the middle letters (only the letters which cover the word).
'on the house' becomes 'free' (something on the house is given away free).
'chimney' with its centre taken out is 'cy'.
'fan'+'cy'+'free'='FANCY-FREE'
(Other definitions for fancy-free that I've seen before include "Lacking responsibility - not currently in love" , "Having no commitments" , "With no emotional ties" .)