Jonatan sends one of his servants to arrange a meeting with the Duchess of Chelam's servants as soon as the Duke of Fraga informs him of the newspaper article. It's lunchtime when he sees the newspaper, and it takes some time for his servant to locate her; he's finished the letter by the time the servant returns.
He is, in fact, angry with her, though he's trying not to let more than a little of it show. But he doesn't, in fact, want to risk Cheliax being brought to ruin by increasingly dubious forms of radical political strategy, and it is theoretically possible that the Duchess has somehow managed not to realize that that's the course she's trying to set them on. He intends to be meticulously reasonable until such time as the Duchess proves impossible to negotiate with.