Sentence:  And Jon—sweet, dull Jon—almost immediately shares his big secret with his sister-cousins, even though it means shattering the appearance of a united front, which is kind of important, since they are about to attempt to <WSD>lay siege</WSD> to King's Landing. 
Meaning of the word marked with <WSD> tags:  To besiege, to engage in a siege; used with "to" or "against".