Approximating the value of k in $k$-mean clustering problem
This problem is known as Geometric Set Cover (which deals with covering with different shapes, so unit balls are a special case). I'm unaware of any relation to $k$-means which is a clustering algorithm whose clusters are not limited in radius.

The problem is known to be NP-complete for $k\geq 2$.

For unit balls, this problem has a PTAS ( ($1+\epsilon$)-approximation) ) for the 2D case and is known to be APX-hard for $d=3$.