 Solution

Let ${\mathfrak{m}}_K$ be the maximal ideal. For all sufficiently large $r$, the convergent p-adic logarithm and p-adic exponential give the principal-unit logarithm isomorphismMultiplication by a power of a uniformizer identifies the additive group ${\mathfrak{m}}_K^r$ with $({\mathcal{O}}_K,+)$. Since $1+{\mathfrak{m}}_K^r$ has finite index in ${\mathcal{O}}_K^{\times }$, it is the required subgroup.