For some reason she imagined little groups of Aes Sedai scattered from the Blight to the Sea of Storms, and constant streams of sisters flowing between, gathering information, sharing information. A peculiar thought. Sisters used eyes-and-ears to watch the world, and rarely shared what they learned unless it was a threat to the Tower itself.
I don't think it's the Black Ajah manipulating the Hall of the Tower, I think it's something like this--a handful of experienced Sisters, maybe the heads of the Ajahs, quietly taking measures to preserve the Tower so that whatever disaster Elaida inevitably leads them to, they can recover.
I have no idea what those measures are, but everything in me recoils at the idea of the Black having such a firm hand in things. Maybe I'm as bad as the Aes Sedai for refusing to see it, but the Black seem too isolated and furtive to be responsible for a plot big enough to include both factions of Sisters.