[For part I, see J. Pure Appl. Algebra 213, No.~6, 1046--1063 (2009; Zbl 1165.18006).] The Pontrjagin dual of a commutative compact group is a commutative discrete group. The Tannaka-Krein dual of a compact group or Hopf algebra is an autonomous (= compact) monoidal linear category which seems to be a structure of a different nature from group or algebra. However, Hopf algebras and autonomous monoidal categories are both examples of autonomous pseudomonoids in appropriately chosen monoidal bicategories. The purpose of this sequence of papers is to extend the theory of Hopf algebras to the context of monoidal bicategories. This second paper places in that context the fact that the Tannaka-Krein dual of the Drinfeld double of a finite-dimensional Hopf algebra is equivalent to the monoidal centre of the Tannaka-Krein dual of the Hopf algebra.