Wrote new play's capital part (5,7)
I believe the answer is:
tower hamlets
'capital part' is the definition.
(borough of London)
'wrote new play's' is the wordplay.
'new' is an anagram indicator.
'play's' becomes 'hamlet's' (play by William Shakespeare).
'wrote' anagrammed gives 'tower'.
'tower'+'hamlets'='TOWER HAMLETS'
(Other definitions for tower hamlets that I've seen before include "Part of London" , "London borough -- Matthew's role (anag)" , "London area?" , "London's docklands borough" .)