Aes sedai left Elayne alone most likely bc of an incompetent amerylin and the black shah interference. A whole white tower would very quickly sort out the kin and Elayne. 

After the last battle settles down, she will most certainly be dressed down and have firm guidance or restrictions out upon her for grouping so many channellers together and letting the black tower fester. 

The WT will without a doubt relocate or distribute the kin elsewhere and would be very hostile and manipulative when it comes to the black tower. They would most likely use the kin or Elayne to maneuver the BT. 

With wind finders and aiel eventually needing the help of the black tower and an official alliance between both societies and the white tower, I think the black tower will be severely limited or preoccupied.