Drafted by who using what qualifications?
I suppose they'd be drafted by whoever was in charge of administering the government department responsible for doing so. We're basically just talking about randomly drawing names from a hat so how hard can that be? As for vetting and confirming slates of drafted representatives these could be put to a fairly conventional election at the riding level. In fact I'd suggest drafting, vetting and electing within ridings could also be randomized so new fresh ideas and perspectives on current events were introduced into the mix. There'd still be plenty for politicians to do if there was always an impending election.
How many people would be able to leave their jobs and careers for four years to serve on this assembly?
Out of millions of potential candidates? Plenty I suspect. Are jurists hard to come by or something, are employers permitted to fire them if they accept the honour/duty? The cases they deal with sometimes drag on for years too so...there are also provisions for jurists opting so there's no reason why that shouldn't be the case for representatives.
Politics and politicians are everywhere, not just in government.
That's fine, they'd still be free to organize themselves and influence people's thinking this way or that as always.