For convenience of notation, let me write the expectation $\mathop{\mathbb{E}}_i t_i$ to denote the average $(\sum_{i=1}^n t_i)/n$.
If I understand your construction correctly, you have disjoint balls of radius $1$ centered at $x_i = \sqrt{2} e_i$ contained in a ball of radius $1+\sqrt{2}$ centered at $y = 0$. This construction, which places $n$ balls tightly packed at the vertices of a regular simplex, is optimal in terms of the positions $x_i$. For the exact optimal bound for your problem, you should pick $y=\mathop{\mathbb{E}}_i x_i$ to get the radius $$\boxed{k_n = 1+\sqrt{2 (1-1/n)}}.$$
The claim that placing the $x_i$ at the vertices of a regular $(n-1)$-simplex and $y$ at the centroid of this simplex is optimal has been proven many times before in many different contexts. For example, it is implied by a bound known by various substrings of "the Welch-Rankin simplex bound" in frame theory. Here's a simple direct proof:
By the triangle inequality, a ball of radius $1+r$ centered at $y$ contains a ball of radius $1$ centered at $x_i$ iff $\lVert x-y\rVert \le r$. Two balls of radius $1$ centered at $x_i$ and $x_j$ are disjoint iff $\lVert x_i - x_j \rVert \ge 2$. Therefore, your problem asks to minimize $1 + \max_i \lVert y-x_i\rVert$ subject to $\min_{i\ne j} \lVert x_i - x_j\rVert \ge 2$.
Working with squared distances is easier. The maximum squared distance $\max_i \lVert y-x_i\rVert^2$ is surely at least the average $\mathop{\mathbb{E}}_i \lVert y-x_i\rVert^2$. This average is minimized when $y$ is itself the average $\mathop{\mathbb{E}}_i x_i$, in which case it equals $\mathop{\mathbb{E}}_i \mathop{\mathbb{E}}_j \lVert x_i-x_j\rVert^2/2$. Each term where $i=j$ contributes $0$ to this expectation, while each term where $i\ne j$ contributes at least $2$, so overall this expectation is at least $2(n-1)/n$. Thus the maximum squared distance $\max_i\lVert y-x_i\rVert^2$ is at least $2(n-1)/n$ and thus $1+r \ge 1+\sqrt{2(n-1)/n}.$ We can check that the optimal configuration mentioned before achieves this bound either by direct calculation or by noting that it achieves equality in every step of our argument.
