To me, Hausdorff metric is an unaccustomed way of making such a space of spaces. I think I don't trust it because fixing a homotopy type gives you a set that is neither closed nor open in general.
But yes I believe the picture is that some kind of "space of spaces of homotopy type XX" is closely related to A(X)A(X).
Let's start with smooth manifolds, but of codimension zero. For a fixed nn and a finite complex KsubsetmathbbRnKsubsetmathbbRn, let Mn(K)Mn(K) be the space of smooth compact nn-manifolds NsubsetmathbbRnNsubsetmathbbRn containing KK in the interior as a deformation retract. (Let's say, the simplicial set where a pp-simplex is a suitable thing in DeltaptimesmathbbRnDeltaptimesmathbbRn such that the projection to DeltapDeltap is a smooth fiber bundle.) You can map Mn(K)toMn+1(K)Mn(K)toMn+1(K) by crossing with [−1,1][−1,1] (and doing something about corners), and you can consider the (homotopy) colimit over nn. Using the classification of hh-cobordisms you can work out that the set of components is the Whitehead group of KK. The loopspace of one component is the smooth stable pseudoisotopy space of KK. To get the idea, think of the case when KK is a point: the space Mn(K)Mn(K) is then, after you discard extraneous components corresponding to cases where the boundary is not simply connected -- which were going to go away anyway upon stabilizing over nn -- your quotient of {embeddings DntomathbbRnDntomathbbRn}~O(n)O(n) by {diffeomorphisms DntoDnDntoDn}. It's also a kind of "space of all hh-cobordisms on Sn−1Sn−1, and thus a delooping of the (unstable) pseudoisotopy space of Sn−1Sn−1.
When KK is more complicated than a point, it's important to distinguish between the space of all blah blah blah containing KK as a deformation retract and the space of all blah blah homotopy equivalent to KK; they differ by the space of homotopy equivalences KtoKKtoK.
There is a similar story for the piecewise linear or topological case.
The piecewise linear manifold version of this construction can, I believe, be shown to be equivalent to a non-manifold construction more like what you asked about: some kind of "space of compact PL spaces in mathbbRnmathbbRn containing a fixed KK as deformation retract".
Waldhausen tells us that the stable smooth construction above and the stable PL construction above are respectively (the underlying spaces of spectra which are) the fiber of a map from the suspension spectrum of KcuppointKcuppoint to A(K)A(K), and the fiber of a map from A(∗)wedge(Kcuppoint)A(∗)wedge(Kcuppoint) to A(K)A(K).
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