When using KNN, how do I know which order of the Minkowski distance to use?
Unfortunately, there is no rule (either general or rule of thumb) for using a given distance rather than another. Minkowski distance of n-th order could work greatly for some task and some datasets, and extremely bad for others.

The reason Euclidean distance is used as default is due to its simplicity and the fact that there is no reason to consider more complex formulae as better a priori.

I suggest you to try multiple measures of distance (or multiple Minkowski orders) and check which works best for your current task. Please consider the risk of overfitting. I would run models more than once, for example using techniques for keeping overfitting under control, such as k-fold Cross Validation, and/or triple train-validation-test splits.