= Solution
A finite <irreducible Markov chain> is positive recurrent. Its <mean recurrence time> to state $i$, starting at $i$ and counting the first positive return, is $1/\pi_i$. This identity comes from the long-run fraction of time spent at that state: successive return cycles each contain one visit at their start, so the fraction is the reciprocal of the mean cycle length. It remains valid for a periodic chain.
Here $\pi_0=2^{-k}$, hence
$$
\boxed{\mathbb E_0[T]=2^k\text{ minutes}.}
$$
For $k=1$ the return is deterministically after two minutes, which checks the formula.
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