We have loads of Kallax! I have 2 boys aged 1 and 4 - you might need to adapt advice for age/interests.
One day, the kids dumped out every single box on the floor, I took it as an opportunity to sort everything into piles/categories.
That meant I could chuck out any broken or worn out stuff.
Sneak out anything outgrown/no longer wanted and put into a donate box.
Locate parts that always migrate away from the toy they need to be used with (put these aside, including the main part)
Any duplicates that weren't useful (e.g. two toy horses = fine, but two incompatible marble runs = frustrating) I got rid of the worst one.
Any categories that were insanely exploding (for us, it's toy cars) I weeded out some, although not many if they could reasonably all be used in a game. I went through all the craft/art stuff and got rid of anything dried up or used up.
Then this is what I'd do.
Large toys (e.g. for us, these are mostly large sized vehicles) go in an empty square of their own, low down and easy to access/put away.
Bord books in low open squares. Easy to get out and put away. Delicate books on higher squares (or in a separate bookcase)
Toys which have specific parts which go with them and are useless without (e.g. shape sorter) or board games and puzzles go into an empty square, but slightly higher than the kids can reach themselves, meaning you only take them down when they ask for help. If they are older, then you could make a rule: These toys need to be put away again when finished with. However, keeping them visible helps because when tidying up, rather than just sling the shapes from the shape sorter into a generic toy box, you see the shape sorter and can put them easily in. Designate a box/open square near the top as a "lost pieces box" so if you find board game or puzzle pieces but can't immediately and easily access the correct box, they go into this box instead. That means they don't immediately go missing again. On discovering a puzzle/game/toy with missing part, check this box first.
Toys which need more supervision (e.g. playdough, crafts, kits) go right at the top perferably inside a cupboard so that they can't immediately be seen. Or if you have stuff to store in there that's off season (outdoor toys) or isn't really a toy (chargers, anything that's not a toy)
For the rest, colourful pull-out drawers with labels stuck on the front with tape saying what should go in them. For us we have:
Food / shopping
Role play / dolls
Cars / trains / vehicles
Farm animals
Musical / noisy
Baby toys (but this is soon to be retired)
Chunky vehicles / Playmobil 123
Construction / blocks / tools
Roads / signs / props
I don't have rules as such about how many drawers can come out at once, but it makes it easier to clean up when everything has a set place, even if it's sometimes ambiguous (is this vehicle chunky or not, does the plastic tree go with animals or road props) - it just helps. And I find they tend to take 1-2 out at a time and then they have a little self-contained module of relevant toys that they can use all together in a game, rather than having a box which has three saucepans, four cows, two dinosaurs, three non-matching pieces of road or fence, some lost pieces from a puzzle, a screwdriver that they both fight over etc - that means they have to get every box out to look for more tools or cooking stuff and it makes a huge mess.
You do have to go through periodically and remove stuff from boxes that isn't meant to be there and return it to the one it's supposed to be, and with the youngest being so little I am fairly often (6 monthly or so) shuffling what is in each box as the older ones become less relevant. But in general, it works and is helpful.
A colourful rug, or a rug with play value e.g. different coloured patches (can be used for all kinds of pretend rooms or zones), hopscotch design, printed road layout etc would also brighten up the room and be soft to sit on.