Peagreene I thought this was all sounding familiar, there was someone who posted with this situation in January, it was the 9 month stage and the person's own manager was on long term sick as well, wonder if that was you.
Don't need to answer, it doesn't actually make any difference obviously, and I won't say who the other person was, it was just all ringing bells!
You say you've done the whole objectives, monitoring thing, but if what he should be doing is showing initiative, taking on projects etc and that's what he's not done, were any of the objectives/monitoring actually about those things? I am assuming you can't move on to a disciplinary process because he managed to achieve the objectives set previously?
So it's a case of starting again. You do have to go through a whole palaver as everyone's said, but you can speed it up by setting the right objectives, being clear about what exactly is required and what he is not doing, which isn't necessarily what's on his job description. If it's the level and type of role where a bit more is required then it's fine to be monitoring that.
If the problem is he is showing no initiative and keeps needing to ask you what to do, that's what to monitor. You have his objectives already, just make it clear that he is expected to achieve them without asking what to do. I'd set a timescale with some 'normal' objectives, but the monitoring should be about how they are achieved, not whether they are (although obviously if they are not that's also something to act upon).
I'd set up a weekly meeting with him for performance management purposes, at which he tells you what he's done. During all of this, make a note every time he asks you something during the working day, so that you have a record of exactly how much help you have been giving him and how many times he is not able to demonstrate the initiative and independence he needs to show. Agree all this at the beginning and be very clear that if there is not a marked improvement in this area of his performance over a reasonable timescale (depends on his job and context, maybe a month, maybe 3, you'll be able to judge) you will then have to consider taking the disciplinary route. Review at the end of the timescale and if you feel improvement is not enough, start disciplinary proceedings
If you are not getting support from HR or your line manager, take the initiative yourself. Just tell them this is what you are proposing to do, have they got any objections or comments which you will be happy to listen to. If/when it gets into disciplinary territory they'll all have to be involved anyway, but at the moment all you need is a 'heads up' about what's coming.
Chances are it won't get to the stage where you have to sack him. I take it from what you've said that he's relatively senior? Very very rare to actually go all the way in that case. IME once someone fairly senior has gone through a couple of months of being 'performance managed', then a couple of months of disciplinary investigation and hearings, then appeals, then a bit more performance management, then another disciplinary resulting ina final warning, then another appeal, then again to be dismissed, etc etc, at some point along the line they get the message, and look for something else/resign before they can be sacked.
Nevertheless, all this is going to be hugely time-consuming for you. It's so important you keep on top of it though, keep notes of everything, make sure he has plenty of opportunity to tell you if there's anything else you should be aware of that's affecting his performance and make sure you have all those meetings. If you are piling on the pressure like this, he is more likely to get the message that maybe this isn't the place/job for him and start looking. It really mustn't be an easy ride for him at all.