I know that ～もくそも（ない) is is an expression that can mean: not at all; not even a little
However I'm still confused at what もクソも in 何かもクソも寝るだけだよ means.!
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This ～もクソも(ない) is a dirty version of ～も何も(ない) described here and here. In slangy speech, 何 in ～も何も can be replaced by クソ, ヘチマ, へったくれ, etc., which are basically metaphors for crappy/meaningless things (see this discussion in Japanese). In this context, he is saying asking "何(か)" to him is meaningless.
What are you doing?
Why d'you ask? I'm just about to go to bed.