Old king holding on in institution (7)
I believe the answer is:
college
'institution' is the definition.
(college is a kind of institution)
'old king holding on' is the wordplay.
'old king' becomes 'cole'.
'holding' is an insertion indicator.
'on' becomes 'leg' (cricket term, fielding positions).
'cole' going around 'leg' is 'COLLEGE'.
'in' is the link.
(Other definitions for college that I've seen before include "prison" , "eg Eton" , "University establishment" , "Educational institution" , "School" .)