Let me just summarize the comment thread in case someone more knowledgeable is willing to intervene.

Artie observes that The Cox Ring of a Del Pezzo surface by Batyrev--Popov shows that every effective class on a del Pezzo with $k \geq 2$ is a sum of $(-1)$-curve classes, with one exception. This means that no effective classes can be indecomposable in the sense above except for those of $(-1)$-curves. All the $(-1)$-curve classes are clearly indecomposable and have $N_\delta = 1$. And of course it is easy to write down these classes explicitly.

The one exception is that if $k = 8$ the anticanonical class is not a sum of two nonzero effective classes. This class is represented by (the strict transforms of) the pencil of cubics through the 8 blown up points. There are 12 singular cubics in the pencil, hence 12 that are rational, and it seems that we should have $N_\delta = 12$ by the OP's definition. The catch is that Kontsevich-Manin "expect" $N_\delta = 1$ for an indecomposable class on a del Pezzo (page 29 of the article in the question, following Claim 5.2.3).

So the question is, have we misunderstood something along the way (more likely, e.g. the definition of $N_\delta$), or did the authors forget a minor case? (A secondary question: is this already worked out somewhere?)