I never used a k-WL in practice, but I did apply weisfeiler-lehman for my graph tasks.
As you can know, the WL provides the coloring by interactive procedure that's assign each node a 'color' (basically some kind of label reflecting the node neighborhood). Counting colors allows to compare two graphs on isomorphism, but it's not that important here, the key is that you have kind of feature for each node.
The k-WL make similar thing, but it operates not on graph, but in space of k-tuples of nodes. Basically, it's pairs or triplets of nodes. Then you define neighborhood as all pairs/triplets that differ in one element from the target pair/triplet. So, now you have have some kind of objects and defined neihgborhood for each of them, that's basically enough to apply the WL coloring.
Ok, now to k-GNN part. You repeat k-WL procedure on pairs of nodes, gets color for each of pair. You use one-hot encoded colors as input features and make graph convolution for pairs (with neighborhood defined as above), then getting some kind of embeddings for pairs. To get the node classification, you make average pooling by all pairs containing the node and then use usual fully-connected neural net.
As of the examples, the paper author provided several in article repo, you can try them.