Tiny part of Jerusalem sheltering sick Thai leaders (9)
I believe the answer is:
zillionth
'tiny part' is the definition.
'jerusalem sheltering sick thai leaders' is the wordplay.
'jerusalem' becomes 'zion' (a name for Jerusalem in the Hebrew Bible).
'sheltering' means one lot of letters goes inside another.
'sick' becomes 'ill' ('ill' can be a synonym of 'sick').
'thai leaders' becomes 'th' (I am not sure about this - if you are sure you should believe this answer much more).
'zion' enclosing 'ill' is 'zillion'.
'zillion'+'th'='ZILLIONTH'
'of' is the link.
(Another definition for zillionth that I've seen is " one coming after many others".)