Balls in Hilbert space

I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real Hilbert space of sufficiently large dimension or infinite dimension contains $n$ pairwise disjoint open balls of radius 1. (The dimension of theHilbert space is irrelevant as long as it is at least $n-1$ since it can be replaced by the affine subspace spanned by the centers of the balls.) We obviously have $k_1=1$ and $k_2=2$, and it is easy to see that $k_3=1+\frac{2}{\sqrt{3}}\approx 2.1547$. The interesting fact is that $k_n\leq 1+\sqrt{2}\approx 2.414$ for all $n$, since in an infinite-dimensional Hilbert space an open ball of this radius contains infinitely many pairwise disjoint open balls of radius 1 [consider balls centered at points of an orthonormal basis]. The obvious questions are: (1) What is $k_n$? This may be known, but looks difficult since it is related to sphere packing. (2) Is $k_n$ even strictly increasing in $n$? (3) Is $k_n<1+\sqrt{2}$ for all $n$, or are they equal for sufficiently large $n$? (4) Is it even true that $\sup_n k_n=1+\sqrt{2}$? It is not even completely obvious that $k_n$ exists for all $n$, i.e. that there is a smallest $k$ for each $n$, but there should be some compactness argument which shows this. I find it interesting that the numbers $1+\frac{2}{\sqrt{3}}$ and $1+\sqrt{2}$ are so close but the behavior of balls is so dramatically different.I suppose the question is also interesting in smaller-dimensional Hilbert spaces: let $k_{n,d}$ be the smallest $k$ such that an open ball of radius $k$ in a Hilbert space of dimension $d$ contains $n$ pairwise disjoint open balls of radius 1. Then $k_{n,d}$ stabilizes at $k_n$ for $d\geq n-1$. What is $k_{n,d}$? (This my be much harder since it is virtually the sphere-packing question if $n>>d$.)