Metrics to determine K in K-cross fold validation
The rule of thumb is the higher K, the better.


I think a better rule of thumb is: The larger your dataset, the less important is $k$.

However, it is useful to have a general understanding of the impact of $k$ on the performance estimator (leaving aside computational costs):


Increasing $k$ decreases the bias because the training set better represents the data
Increasing $k$ increases the variance of the estimator because the training data sets are becoming more similar


Also note that there is no unbiased estimator for the variance of the $k$-fold CV. Together this means that there is no metric that can tell you the best $k$ if you leave computational costs aside. Some empirical studies suggest that 10 is a reasonable default. 

And to be clear, $k$ is not a hyper-parameter you want to tune to find the best accuracy. If you start performing $k_2$-fold CV to find the best $k_1$ something hopefully feels wrong.