If you don't have the patience to slowly fill the void with spray foam through holes drilled in one of the studs, you could (if you were an experienced carpenter) use a Sawzall to remove one of the studs and re-frame the corner as a two-stud corner. Then you could either insulate the void with a fiberglass batt or simply present the two-stud corner to the spray-foam contractor as an accessible corner.
Spray polyurethane foam, or SPF, is the main type of 2-part, closed-cell spray foam used by insulation contractors. Large-scale insulation jobs require special equipment as well as safety gear to protect the installer from chemical fumes during installation. When the foam cures and hardens just several minutes after application, it's completely safe for as long as it stays in place.
Spray polyurethan foam does effectly prevent all six of these heat transfer methods from occurring. These should be the mechanisms by which you judge insulation, in addition to their R-Value. If you're uncertain about installing spray foam insulation, Energy Star has a map that recommends levels of insulation by R-value and cost-effectiveness in the United States.
I recently did a remodel project for my basement using spray foam insulation. The original builder has used fiberglass, but I wanted the best insulation i could find. We chose closed cell foam because it blocks moisture from getting into the house. It took one day to install and was fumey for about 3 hours. It was pretty cool watching it being done. It comes out as a liquid and then expands rapidly into a foam. It's quick! Then they took a types of saw to shave it flat with the wall studs so we could then go ahead and drywall.