Lady Blakeney is so desperate for the identity of the Scarlet Pimpernel to become uncovered because her brother's life depends on it. Earlier on in the novel, Chauvelin managed to obtain a letter written by Lady Blakeney's brother, Armand, which clearly links him to the Scarlet Pimpernel and the others who are supporting the French aristocracy as they escape. Chauvelin has used the letter as blackmail in order to try and trap the Scarlet Pimpernel and identify who he actually is. In return for Lady Blakeney's help, he has promised that he will spare her brother. This is why, in Chapter 15, Lady Blakeney is so eager that Chauvelin's trap will work, even though she feels pity for the Scarlet Pimpernel:
Woman-like, she thought of him with unmixed sadness; the irony of that fate seemed so cruel which allowed the fearless lion to succumb to the gnawing of a rat! Ah! had Armand's life not been at stake!...
What drives Lady Blakeney to such agitation therefore is the fear that she faces regarding her brother and the realisation that his safety depends entirely on the capture of another man. Her distaste for this situation and also her personal liking of the Scarlet Pimpernel is shown in this quote, that clearly states the only reason she is participating in such a scheme is because she is forced to.
Chauvelin has intercepted papers that connect Marguerite's brother, Armand, to the Scarlet Pimpernel and others who are aiding the nobility of France in their escapes from the country. In return for her brother's safety, he tells he as she watches the opera Orpheus that if she helps him discover the identity of the Scarlet Pimpernel, he will ensure the life of her brother.
"She would not allow herself any more time to think. Her early, somewhat Bohemian training had made her something of a fatalist. She felt that events would shape themselves, that the directing of them was not in her hands. From Chauvelin she knew that she could expect no mercy. He had set a price on Armand's head, and left it to her to pay or not, as she chose."(Chapter 12)
"That letter of Armand's -- foolish, imprudent Armand -- was in Chauvelin's hands. Marguerite knew that as if she had seen the letter with her own eyes; and Chauvelin would hold that letter for purposes of his own, until it suited him to destroy it or to make use of it against Armand. All that she knew, and yet she continued to laugh more gaily, more loudly than she had done before."(Chapter 10)