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

Artie observes that <a href="http://arxiv.org/abs/math/0309111">The Cox Ring of a Del Pezzo surface</a> 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 <a href="http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf">the article</a> 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?)