This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\varepsilon$ and $σ$. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of $f$ and $A$. This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method. We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for $f$ and $A$ and noise levels $\varepsilon$ and $σ$. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.