I would not expect him to get a job in a month. Instead you could agree a number of jobs to apply for. My DS has applied for internships and failed to get them, and each application was a heck of a lot of work. And when my DH was made redundant (it's happened a couple of times) applying for work was a full-time job in itself. At 17 your DS may not be able to get full-time work, and I don't even know if he'd be allowed to work in a chipshop at that age for heath & safety reasons. Any kind of job, zero hours, part time, whatever, is a start; but applying for jobs is the real starting point.
If he does not get a paid job after a month or so and isn't in college then he needs to look for some volunteer work instead. Charity shop, sports tutor, whatever. But don't talk to him about that yet. One thing at a time. First he should be looking for paid jobs and applying - if he has dropped out of college, that is. It's not clear to me that he has.
I am not clear at what point he can sign on, everything has changed so much since I was young. When I left school at 18 my mother said I had to sign on and that motivated me to find a job within a fortnight! But things have chnaged so much, I don't even know if a 17 year old can sign on. But as soon as he can sign on then if he doesn't have a job and isn't at school then that should be another condition.
the day to day sticking point is the tidying/chores. This is the one which 24/7 has most effect on me even though it's not per se as important as his mental health, future, etc. Someone said he should be given some 'responsibilities' to deal with maybe in a specific time frame rather than immediately I want it done. Ok...ish… But how on earth do I rein in the sheer frustration, disappointment and anger when he (cos I fear he will) leaves a mess in the kitchen, or shoes all over the hall, shambolic sofa with drinks, X box controllers strewn around, etc etc, which has left me doing 98% of everything for two years-plus? Literally HOW?
A couple of suggestions .... As wll as some regular chores that he controls, can you create zones? Zones like his own room that he can leave as much in a pigsty as he likes? And zones of your own that he can't touch? He can't be in the living room in the daytime if he leaves it a mess. He can use the Xbox in his own room only. I get the impression there's a lack of boundaries between you and him, maybe it's a result of grief but everything little he does over-affects you, and everything you expect over-burdens him. So just some zones and personal space could help.
Also mimimal-effort ways to tidy the shared areas, ways that affect him more than you. Get a big tub, if his shoes are on the hall floor they get chucked in the tub and the tub dumped in his room. His stuff stays in the tub til he takes it out (either in a heap on his bedroom floor or properly put away, his choice). You don't have to look after his stuff better than he does.
One other suggestion, is simple consequences. Not huge lifestyle ones but small immediate logical ones. He's leaving a mess in the kitchen? You don't cook for him in a messy kitchen. If the kitchen is messy then you cook for yourself only that night. And leave his mess lying.
I'm sure things will better 
. Small steps.