How can I thaw a frozen exterior drain quickly?
You can pick up a cheap water pump that you can use to remove the water.  If you have a Harbor Freight in your area, they have a good selection of cheap pumps.  If your drain is metal, you can use a brush torch with a 15 pound propane tank to help remove the ice.  Use it to help break up the ice.  It will probably not be efficient to melt all of the ice.  Warm water may help break up the frozen water in the drain pipe.