Demonstrate king, for one, is especially fine specimen (9)
I believe the answer is:
showpiece
'fine specimen' is the definition.
'demonstrate king for one' is the wordplay.
'demonstrate' becomes 'show' (synonyms).
'king for one' becomes 'piece' (king is a kind of piece).
'show'+'piece'='SHOWPIECE'
'is especially' is the link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for showpiece that I've seen before include "shining example" , "jewel" , "Outstanding item" , "Impressive exhibit" , "fine specimen of its kind" .)