Just what the tower represents. She may not have spent much time in the tower itself, but ever since she got the idea that she could be an Aes Sedai, that's been one of her major goals. She may have had other things on her mind through out the books, like helping Rand, or getting trained with the Aiel, but even then, she's always had this idea of being Aes Sedai and upholding what it means to be Aes Sedai in the back of her mind. And this attachment to Aes Sedai only grew when she ended up as the Amyrlin seat and started feeling like it was basically her responsibility to bring them back together. Seeing how broken the people in the Tower were strengthened that thought. So it doesn't really matter how long she was in the Tower itself, its more that she feels connected, now, to Aes Sedai in general, and hates seeing this once powerful group falling apart.