= Solution
Take $\alpha=I/(i\hbar T)$ in the supplied Poisson summation identity, initially with a convergent regulator. Its left side is the winding sum. Multiplying its right side by the prefactor in part d gives
$$
\sqrt{\frac{I}{2\pi i\hbar T}}\sqrt{\frac1{2\pi\alpha}}=\frac1{2\pi},
\qquad
-\frac{q^2}{2\alpha}=-\frac{i\hbar Tq^2}{2I}.
$$
Consequently
$$
K_\theta=\frac1{2\pi}\sum_qe^{-iq\Delta-i\hbar q^2T/(2I)}.
$$
Relabelling $q=-l$ gives exactly the spectral answer. The square-root branch is fixed by continuation from positive imaginary-time damping; the limiting kernel has the required periodic delta-function normalization.
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