For $L/K$, any totally ramified cyclic extension of degree $p^2$ of local fields which are finite extensions of the field of $p$-adic numbers, we describe the ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-module structure of each fractional ideal of $L$ explicitly in terms of the $4p+1$ indecomposable ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.