The instrument of badness has no end (4)
I believe the answer is:
bass
'the instrument' is the definition.
(bass is a kind of instrument)
'badness has no end' is the wordplay.
'has no' indicates named letters should be taken away.
'badness' with 'end' removed is 'BASS'.
'of' acts as a link.
(Other definitions for bass that I've seen before include "Freshwater fish and lowest voice" , "bottom of sound?" , "Fish, guitar" , "Fish for the low-voiced" , "Lowest adult male singing voice" .)