  We provide an explicit resolution of the existence problem for extremal
Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More
precisely we show that M admits such a metric if and only if its rational
Delzant polytope (which is a labelled quadrilateral) is K-polystable in the
relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi
et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric,
i.e., compatible with a conformally equivalent, oppositely oriented toric
Kaehler metric, which turns out to be extremal as well. These results provide a
computational test for the K-stability of labelled quadrilaterals.
  Extremal ambitoric structures were classified locally in Part I of this work,
but herein we only use the straightforward fact that explicit Kaehler metrics
obtained there are extremal, and the identification of Bach-flat (conformally
Einstein) examples among them. Using our global results, the latter yield
countably infinite families of compact toric Bach-flat Kaehler orbifolds,
including examples which are globally conformally Einstein, and examples which
are conformal to complete smooth Einstein metrics on an open subset, thus
extending the work of many authors.
