Two Face is usually depicted with such a clear split personality that I can't see him not going to Arkham. Nolan's version would not qualify though. Ventriliquist suffers from the same condition. Jervis Tetch (Had Mtter) and Maxie Zeus would also qualify, Tetch because he actually hears voices in his head and Zeus because his delusions truly make him think the law doesn't apply to him.
Outside of those 3 I don't think anyone qualifies. Poison Ivy would if she really couldn't talk to plants, but since she can it's not a delusion or voices in her head. Harley Quinn is another borderline case but I lean towards no.
The others may have specific issues like psychopathy, narcisism, obsessive-compulsiveness-disorder, depression, ect, but they are not legally insane.