How many cards do I need to make a collected set "draftable?"
It depends on how you want to build your packs. If you plan to build proper packs then the choices are different then if you just want to grab random cards. The best experience would be from the former so I will go off it.

The double faced cards were seeded from a sheet with a single mythic, two of each rare, six of each uncommon and eleven and each common. Although you could reduce the card count by half if you wanted without killing the chances too much (more Garruk and higher or lower common chance to taste). Pretty straight forward to reproduce for either 60 or 121 cards (mostly common).

Your normal rare and mythic slot would be best served by one of each mythic and two of each rare, but you can design a system to reproduce the 1/8 ratio (remember there are 4 rares per mythic) such as using half the mythic cards you have before shuffling the rares.

Going to common and uncommon you probably want to drastically cut the number of copies. I would think two of each uncommon would be enough, although you could do three if you wanted that possibility. Similarly with commons four to six would work depending on how many dupes you want to see.

For simplicity I would either pick 1:1:2:4 or 1:1:3:5 for each card type and handle the various slots separately. Although once you have built this out (or have a tool to test) you could try just building normal packs from either ratio and see what rarity density you see.

Finally if you wanted really good packs it may make sense to pre-seed each pack with a common of each color (or two cards of each color if not segregating by rarity). While that doesn't eliminate packs that suck for some people it does stop the "nothing in my color first pick" annoyance in cubes.