Thanks for all the great and funny advice, guys. It's a complicated situation that I can't really explain fully without writing up pages and pages about it. So complicated I barely even get it myself. On the one hand, he's on the chopping block. On the other hand, my worst fear is that he will get away with it for years here like he did at the last place. I have definitely gone the "document it and report it" route for a long time with him till it got to the point where the last company told me they would never fire him so I just quit trying. When they finally did fire him (for something completely unrelated to his quality of work), everything there got better.
When it comes to assigning him tasks, his job is one of information gatherer and client liaison. All I need from him is to actually gather the information, which he really stinks at. The reason he's often viewed as "irreplaceable" is because he has the unique skill to bully angry customers into submission. The bosses have always been afraid to move someone out of that position because it would mean that clients would have to be moved to another representative, which doesn't seem so bad to me but has always caused executives to shudder.
I feel like it's just hopeless. I guess I'm looking for the magic words to say to him to get him to actually do even just one thing that I ask of him, but I know no such words exist because it's not within his capability to be a competent employee. So maybe I'm just venting.