Ambitoric geometry II: extremal toric surfaces and Einstein 4-orbifolds

We provide an explicit resolution of the existence problem for extremal Kähler metrics on toric 4-orbifolds M with second Betti number b_2(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. Székelyhidi et al.). Furthermore, in this case, the extremal Kähler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kähler 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 Kähler 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 Kähler 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.