Suicide = Self Murder = Murder
There's no difference if you murder someone else, or if you murder yourself.
If you consider murder to be a sin, then you should also consider suicide to be a sin.
Hashem gave people freedom of choice, and the ability to choose good over evil. In the example given in the question, the suicide victim made a deadly choice to kill himself. Ultimately, the decision was his, and the responsibility for his death remains his.
That being said, the person that egged him on to do the dastardly deed, would be guilty of violating several Torah prohibitions. To the best of my knowledge, these prohibitions do not carry any punishment by Beis Din. However they carry a punishment by Hashem in Heaven.
- מסית לדבר עבירה, incitement to sin.
[See Iggros Moshe (OC 1:99) that even though Beis Din only punishes a person who incites to idolatry, the prohibition against "inciting", applies to all sins, and in any case he is punished by the laws of heaven. See also ibid. (OC 2:7) for further discussion of this point].
- לפני עור לא תתן מכשול, placing a stumbling block before a blind person.
Included in this prohibition, is giving bad advice to another person.
The worst "bad advice" that you can give a person, is "advice" to kill themselves.
- לא תעמוד על דם רעך, You shall not stand by [the shedding of] your fellow's blood.
Rashi interprets: "To see his death and you can save him, such as drowning in a river and an animal or a bandits come upon him".
In this case, the woman had the power to save this man from murdering himself, and not only did she not save him, she was part of the cause of his death.
- הוכח תוכיח את עמיתך, rebuking a person who is doing a sin.
"You shall surely rebuke your fellow, but you shall not bear a sin on his account."
[See Rambam (Sefer Hamitzvos, Assei, 205) "The 205th mitzvah is that we are commanded to admonish a person who is performing a transgression or who is preparing to do so. One must verbally warn him and admonish him..."]
In this case, the woman would be obligated to admonish the victim, and deter him from killing himself. Instead, she incited him to sin; the antithesis of rebuke.