Collection is sth new for us. I have the impression that this is permanent addition to the SR gacha. But I'm not entirely sure.
To get Kurito's cards is pure luck based. Every item that you feed her has some value, that increases your chances for getting sth better.
Most people feed her 1 R-gacha or 3 of those. There are also some other combinations.
Here is a thread with a list of combinations that were tried and what the putcome was. But it's a bit outdated, because there are now some new items, so you should also look up some of the newest posts (last 2-3 pages).
http://harem-battle.club/peropero-se...itos-nest.html