The reason they kidnap people like Hassan is to attract greater attention to their cause. Just look at how much attention they got. In a way they do help their cause, because many people understand that she would not have been kidnapped had the US not invaded. People may see no justification in the group’s action, but they do see the effect (even saintly people being murdered) to the cause (US invasion).
If all Americans understood that an offensive attack on Iraq would lead to their favorite people in the US being killed, would they be as likely to support the attack?
Hassan is an involuntary martyr to the cause of Iraqi freedom from US rule. FWIW she was Muslim; she converted.
A similar example is the Beslan school massacre. Those terrorists say they did it because Russia carpet-bombed Grozny, killing hundreds of kids, which did happen. Do you think Russians will be less likely to support carpet bombing in the future? You betcha, even as they vow revenge.