  We investigate Hamiltonian aspects of the integro-differential kinetic
equation for dense soliton gas which results as a thermodynamic limit of the
Whitham equations. Under a delta-functional ansatz, the kinetic equation
reduces to a non-diagonalisable system of hydrodynamic type whose matrix
consists of several $2\times 2$ Jordan blocks. We demonstrate that the
resulting system possesses local Hamiltonian structures of
differential-geometric type, for all standard two-soliton interaction kernels
(KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases). In the
hard-rod case, we show that the continuum limit of these structures provides a
local multi-Hamiltonian formulation of the full kinetic equation.
