I think, as another poster stated, that nurse being on your unit would be a hinderance to bringing on staff nurses worth their salt. You also would probably get more cooperation from your existing staff. You did say that they avoid working the shift before him and going on call when he's scheduled. This is not a can't get your a** moving/few minutes late to a shift, nurse. This guy only shows up when he wants! I (and probably every other working adult in the world) wish I could do that.
Since you're already relying on agency for the time being, at least make it a place they will want to come to work. At this point three agency nurses vs. two does not seem like much of a difference. They would at least be there. Also, if things begin to turn around the agency nurses may be willing to come back on a regular basis and you would have some continuity of care.
This guy is dead wood. He's not documenting patient care?? He's late 85%
of the time?! He's putting the institution's a** and yours
on the line every time he does notdocument care or is really late. What if he messes something up and it's not charted or your staffing ratio while you're waiting for him is pi** poor and someone sues. Everyone's name (including the nurse manager's) goes on the lawsuit.
Since you have spoken to him four times since December, (I assume you documented these.) I would put him on final written warning. I'm sure he'll see the writing on the wall and you'll have the paper to back you up when he inevitably and blatantly disregards it like he has everything else.