Expanding Uniformity

Fix positive integers $m < k$. Andrea has a fair $m-$sided die with the values $1-m$ on it. She wants to generate a uniformly random number from $\{1,2,\dots, k\}$ using this fair die. Once she uses the outcome of a given roll to attempt to generate the uniformly random number from $\{1,2,\dots, k\}$, the roll can't be used in any other comparison. What should Andrea's strategy be to minimize the expected number of rolls that she has to perform? Under this strategy, what is the expected number of rolls she must perform when $m = 5$ and $k = 15$?