How many Great Shinobi Nations, each led by a Kage, are there in Naruto?

How to get close to this number

The elements give it away. This world is organised around fire, wind, lightning, earth and water — a classical five-element scheme that recurs across East Asian cosmology and appears constantly in fiction built on it. If you recognise the scheme, you have the count before you know anything about the series.

The trap is that the story features many more villages and countries than the great ones. Smaller nations appear throughout, and counting locations rather than great powers would give a much larger number. The question specifies nations led by a Kage precisely because that title is what defines membership of the top tier.

The structure is doing narrative work rather than decorating the setting. Five roughly matched powers with shifting alliances is a setup that generates political plot indefinitely — enough actors for intrigue, few enough for the audience to track. It is the same reason five is a common number for great-power systems in fiction and in historical writing about them.