Even though the speed of the electrons is nowhere near the speed of light, electromagnetism is so strong that the tiny length contraction at those speeds is enough to produce the magnetic force.
Yes, this is the correct answer. It's a relativistic effect that is nonetheless observable at velocities not usually considered to be relativistic.
I think this is extremely cool, BTW: any ordinary refrigerator magnet serves to demonstrate the truth of special relativity. Indeed, since the propagation of a photon involves oscillating electric and magnetic fields, the very existence of light itself depends on relativity.
So in a strange and unexpected way, one could say that when Michelson and Morley went searching for the ether, they actually found it. The fact that the speed of light is the same in every inertial reference frame is precisely what allows it to propagate at all.
I may be confusing propagation speed and drift velocity--it's been a while since I learned this, and I think there is some way to rationalize that the net current is really moving close to the speed of light when you add/subtract the motions of all the individual particles, even if none of them in isolation are moving anywhere close to that speed.
In any case, this article explains the whole matter a lot better than I could.
The drift speed of electrons in an conductor is slow. The actual velocity they travel at is much faster. Though the actual range defies me at this point, so I have no idea if relativistic effects are noticeable.