Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
I spend many hours in the bookshops along this stretch, namely Henry Pordes for second hand books and new discounted titles (pile on left of photo), as well as Koenig Books for new art books with lots of great discounts in the basement. (photo right). In 2001 the rents for much of this strip were increased to align them closer to market value but thanks to public support they were reduced again. Unfortunately some bookshops still closed down; Koenig replaced the beloved Zwemmers - a real institution - but thankfully still provide great books, although others are no longer bookshops at all.