Disrespectful library oddly ignored event including monarch (10)
I believe the answer is:
irreverent
'disrespectful' is the definition.
('irreverent' is associated in meaning with 'disrespectful')
'library oddly ignored event including monarch' is the wordplay.
'oddly ignored' means one should take alternating letters.
'including' means one lot of letters goes inside another.
'monarch' becomes 'er' (abbreviation for Elizabeth Regina).
The alternate letters of 'library' are 'irr'.
'event' placed around 'er' is 'everent'.
'irr'+'everent'='IRREVERENT'
(Other definitions for irreverent that I've seen before include "Cheeky" , "Not respectful" , "Disrespectful" , "Impious" .)