I can understand situations with staffing where losing the person means losing the position, and no manager of a department wants to give up a position. But as a manager, your time is being consumed and YOUR effectiveness is being compromised by this defective employer.
My suggestion is that you carefully analyze what you have going on here. I know there are some wonderful experienced HR professionals here who can offer a lot of suggestions, but here are a few of mine.
What exactly, though, do you mean by "decipher his time"? I spent many years watching people in the private sector try to terminate people for productivity issues. I also worked with many people who were "untouchable" in the civil service sector who turned out perhaps one eighth of the work that others in the same position were producing, (try managing around that) It's bad for morale, bad for your nerves, bad for the overall workplace.
But there are some constructive things I'm sure you've tried or you might try. One question that comes quickly to mind is, is this employee incapable, or is he just a slacker? He is "unable to complete his daily paperwork" you say. As a supervisor, you have to determine if is this because of the amount of paperwork, his failure of competence, in that he just doesn't know how to do it, or is he deliberately slacking, spending too much time away from his desk, making careless mistakes, etc. A program of progressive discipline comes to my mind, in which you try to figure out exactly why this person (after two years) isn't up to snuff and either encourage him to improve that performance or let him "fire himself."
If he is actually terminated for poor performance, the unemployment system would first look at whether or not he was adequately trained, then whether or not the work load is reasonable to expect of him. A VERY important question is "Was there ever a time when this employee was meeting performance standards?" That would indicate that he is capable of doing better but is voluntarily choosing not to do what he is asked to do by his employer with the accuracy that is asked for. That's a real good firing offense.
The unemployment system, (filing for and perhaps being approved for unemployment is the only recourse this employee would have in termination ) would try to determine if the employee was just slow and stupid and really unable to do the work, though he showed up every day and tried his best, or if he is a goof-off who's passing work on to others by his failure to meet performance standards. Or maybe he's just passive aggressive and enjoys aggravating the hell out of his manager, you. I would suspect that instead of outright firing this person right away, that if you have not already done so, you begin a serious, well thought out program of progressive discipline for performance issues, at the same time you have involved your HR in the situation. Your employee might feel that he was being harassed and self terminate, and perhaps you'd be able to replace him. But if no improvement is forthcoming, you can spend more time and perhaps have cooperation from your management in replacing this employee with a more competent one.