The isometry $Y:\mathcal H\rightarrow \mathcal H_E \otimes \mathcal H$ is
$$
Y=\left(\begin{array}{c} A_1 \\ \vdots \\ A_K
\end{array}\right)
= \sum_k |k\rangle \otimes A_k\ .
$$
Clearly,
$$
\mathrm{tr}_E(Y\rho Y^\dagger) = \sum_{kl} \mathrm{tr}(|k\rangle\langle l|) A_k\rho A_l^\dagger =
\sum_k A_k \rho A_k^\dagger \ ,
$$
as desired.
Moreover, $Y$ is an isometry, $Y^\dagger Y=I$, i.e., its columns are orthonormal, which follows from the condition $\sum_k A_k^\dagger A_k=I$ (i.e., the map is trace preserving).

Now if you want to obtain a unitary which acts on $|0\rangle\langle 0|\otimes \rho$ the same way $Y$ acts on $\rho$, you have to extend the matrix $Y$ to a unitary by adding orthogonal column vectors. For instance, you can pick linearly independent vectors from your favorite basis and orthonormalize. (Clearly, $U$ is highly non-unique, as its action on environment states other than $|0\rangle$ is not well defined.)

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