Update: Palpatine could very well have used his powers to help himself, but he was so utterly consumed with rage at Vader that he couldn't focus his energies.
When at last he reaches the open shaft he hurls the Emperor down into
the reactor. It’s a long, long way down and the Emperor fires his
lightning upward as he falls.
He might still have used his powers to save himself, but his hate is
now so strong his only thought is to cause Vader more pain. So the
lightning continues to flicker and flash even after the body is out of
And then comes a great explosion when his body finally reaches the
reactor and a poisonous wind races up the shaft, knocking Vader at
last to the floor. - Return of the Jedi: Beware the Power of the Dark Side
Palpatine wasn't injured (per se) but it's seen in Revenge of the Sith that using Force Lightning for any period of time is an extremely energy-consuming activity. That combined with the fact that he impacted the sides on the way down, most likely rendered him unconscious before he had much chance to use the Force in any meaningful way.
Palpatine’s body, still spewing bolts of light, spun out of control,
into the void, bouncing back and forth off the sides of the shaft as
it fell. It disappeared at last; but then, a few seconds later, a
distant explosion could be heard, far down at the core. A rush of air
billowed out the shaft, into the throne room. Star Wars : Return of the Jedi Novelisation
If you watch the footage in the film, it also looks like some of the lightning hit Vader (who was holding onto Palpatine at the time) so there's a solid possibility that he was also electrocuting himself:
As to the specific question of why Palpatine didn't use a lightsaber to save himself, this answer strongly suggests that Palpatine doesn't actually appear to have a lightsaber on him at the time he fell. Luke's had been "hurled away", Vader's had fallen in the pit and he isn't wearing one of his own.