Niece slept badly, getting sickness (10)
I believe the answer is:
pestilence
'sickness' is the definition.
(pestilence is a kind of sickness)
'niece slept badly' is the wordplay.
'badly' indicates anagramming the letters.
'niece'+'slept'='nieceslept'
'nieceslept' is an anagram of 'PESTILENCE'.
'getting' is the link.
(Other definitions for pestilence that I've seen before include "Plague - lent pieces (anag)" , "dire horseman?" , "Fatal epidemic disease" , "that'll plague you" , "Epidemic disease with a high death rate" .)