Why nine days
Nine days is not a round number, and that is deliberate.
A week is too short. Two of its days are a weekend, and the first is spent deciding what the week is about — so a seven-day sprint is really a four-day sprint with a planning meeting attached.
A month is too long. A month has room in it to be avoided. Anything you can start on day nineteen is something you will start on day nineteen, and the first two weeks quietly become preparation.
Nine days has neither problem. It spans two weekends, so there is nowhere to hide and also somewhere to think. It is short enough that the end is visible from the beginning — you can hold the whole thing in your head on day one — and long enough that a real thing has to move for it to have counted.
One goal. Not a backlog, not a quarter's roadmap. One measurable thing, with a number attached, chosen because it is the thing most worth moving right now. Everything else is explicitly not this sprint's problem.
The number matters more than it looks. A goal without one is a mood, and a mood cannot be missed — which means it also cannot be hit.
Long enough to be wrong in a useful way
The strongest argument for nine days is not that it is the perfect length. It is that it is long enough for the choice of goal to be tested, and short enough that being wrong is cheap.
A month-long goal that turns out to be the wrong goal costs a month. A nine-day one costs nine days and tells you the same thing. Given that picking the right thing to chase is the harder half of the problem, the interval should be sized for how often you want to revisit that decision — not for how much work fits inside it.
Nine days is how often a company should be allowed to change its mind.