Bulk LEGO as presents
Sorting random kits out of a bulk lot and building them can be fun, but it's often quite tricky to determine what you actually have. Also, unless you did quite a bit of sorting up front, you might find that it gets harder and harder to pick out a full kit as time goes on. You could let the kids figure it out, but probably not until they're older, and can look up inventories on bricklink. You could also make it a game: you identify a set candidate, the kids help you build it by finding parts from the lot, and if you succeed in building it you give it to them to keep. If it's a really close build, using bricklink to fill in the missing pieces would make a lot of sense. I once tried to build every single Lego kit my brother and I owned as kids and made a list of all the missing parts. I never got around to it, but a single Bricklink order might be able to clear out that list.

If, instead of sorting out official kits, you sort out the parts by type/colour, you could probably make up some nice little lots of parts that are useful for building. Maybe a bag full of spaceship parts, or castle parts, or car parts. Each of these could become a sort of "kit" on its own, and you could gift them to your children whenever you feel like it.

You could also make some little creations, or large creations, and treat them as "kits" as well. It might be possible for you to recreate the Monthly Mini Models that the Lego company gives away, or small Creator sets, or other similar kits. 

Alternatively, just spread the whole thing on the floor and tell them "Build! Share! Tidy Up After!" and let them go crazy with it.