Is ill here a noun, and thus the object of speak; is it an adjective, or an adverb modifying speak?
It can be understood either as a noun or as an adverb. Merriam-Webster lists this usage under both adverb and noun in its definition of ill:
adv. 1 c : so as to reflect unfavorably < spoke ill of the neighbors >
noun 3: something that reflects unfavorably < spoke no ill of him >
The Free Dictionary similarly shows it as both:
noun 7. an unfavorable opinion or statement: I can speak no ill of her. (TFD)
adverb 2. ill - unfavorably or with disapproval; "tried not to speak ill of the dead"; "thought badly of him for his lack of concern"