I got a little tool that looks like a small pair of pliers, except that its head is two blades that come together for snipping wires and the like. The blades do not run in a plane parallel to the handles, but perpendicular. Using these I'll just grip the blades on to the base of the spike remnant. When I squeeze they start digging in to the metal of the spike itself, and I'm careful not to grip too hard and cut off the remaining part of the spike. When it is dug somewhere in to the spike I then just turn the handle and twist the spikes out.
They haven't failed on me yet, and I've used them when nothing else would work. I don't know, however, if you have enough of the spike left to grip.
These tools should be available at most hardware stores. They are a variant of wire snippers.