I suppose to "flirt with" requires two people to participate. To "flirt at" is flirting unreciprocated.

"Flirt with" is very common. It implies that someone gives a another person signs that they are interested romantically. "Flirt at" would imply that she is flirting, but the person she is trying to get interested in her does not return the feeling. The flirtation is only in one direction. You could also say "flirt with" if the other person is not interested. However, Irene wants to make the point strongly.