The 'Elbow method' [1], yeah. Sweep a bunch of different k values and pick the one where your target metric starts having diminishing returns. There are a lot of heuristics you could use.

There's still not one particularly great way to pick k for this type of clustering. There kind of can't be as clustering is sort of inherently subjective.

Other methods like HDBSCAN can find the number of clusters themselves. There's still a 'one key parameter' you have to pick yourself, but it can be more intuitive and less sensitive for some datasets, when you don't know how many clusters there are / how many you want.

Sometimes, picking k exactly is useful for practical reasons too. Like fixing the size of your codebook. Or partitioning a dataset into k equal-ish pieces, say to spread across a pool of servers of a known/fixed size.

[1] https://en.wikipedia.org/wiki/Elbow_method_(clustering)

You could use X-means instead (https://www.cs.cmu.edu/~dpelleg/download/xmeans.pdf). The idea is to use an information criterion that trades the number of parameters against the reduction in within-cluster variance. The approach essentially splits a cluster when the BIC of the split is larger than without. I’d use this to select k and still run normal k-means though.

There isn’t a well defined correct or ideal answer. It depends on what you’re doing with the clusters. Sometimes the raw data are too numerous to properly deal with and you just want a smaller set of points to reason about. In that case pick the number of points that are convenient to reason about. Sometimes you have a fixed budget for the cluster representation and that defines k. Sometimes just pick sqrt(N) so the average cluster size is k.