Average step rise is 6-8 inches.
If you're young and not tired, 3 steps per second. As you get winded, it decreases to 2.5 steps/second. You get winded more quickly if the rises are higher. However, the question specifies that you lack the energy to run, so let's slow those down a bit to more of a walk or trudge.
An eight-inch rise, 2 steps per second, and you finish the climb in about 5 minutes...but you will be winded and your thighs will complain.
Alternately, a six-inch rise and 2 steps per second, and you finish the climb in a bit under 7 minutes. Hope you remembered a bottle of water!