The precise number of feet traveled probably isn't important, so handwave it and move on.
The exact number of feet that a character or group would have to travel to ascend 100 feet via spiral staircase is going to require a bunch of math, and calculating that number will probably slow down play in a session. In addition, it's probably not going to make a huge difference in play if you just handwave the number; that is, it's probably not going to hurt your players' feeling of verisimilitude that the number is off by a bit. It's important to know how many feet your players will have to travel to go between floors, but it's not important for that number to precisely match reality.
With all that said, I have two ways to handwave this number to present.
Version 1: 200 feet.
One step on a staircase is about 1 foot long. I estimate this because I'm personally a size 11 shoe, and most steps are about the size of my foot. One step up is about 6 inches. I estimate this in the same way; most steps that I travel on are about half the size of my foot. With these two numbers, we know that we have 200 steps to travel (100 feet up * 2 steps per foot) and each step is about a foot long. Thus, 200 feet to travel. This would take 7 rounds to travel, and would have 10 feet of movement left over.
Version 2: 100 feet.
This is the version I prefer. In this case, no calculation is done. It's simply assumed that 100 feet is 100 feet, and that play time is too valuable to spend on calculating the sides of a triangle. In this case, it would take 4 rounds to ascend the stairs, with 20 feet left over.
In either case, I would strongly suggest treating the stairs as difficult terrain. Walking up or down stairs is harder than walking on flat ground, and that's exactly what difficult terrain was meant to represent. If you treated the stairs as difficult terrain, then the two versions would take 14 and 7 rounds, respectively.