Hoo-boy! What excitement at your house today!
First, I don't like to let kitties kitten wherever. I feel it is necessary for me to have access at any time and I don't want her putting kittens out of reach and/or in places I can't keep cleaned and sanitary. For me, this means NO under furniture access.
I like to place them in a large bathroom. Mamma cats really don't need much room to stretch their legs as all they will do is lay with their kittens the first few weeks; only stirring to eat, drink and eliminate. So, bathroom w/ food, water, litterbox.
I like to use a large carrier or a 2'x2' box so it gives the Mamma plenty of room to move around to clean and/or get herself comfortable without fear of laying on any kittens in cramped quarters. I also prefer kitten nests that have a top over it to create a 'private cave' for the mamma and kittens. With a carrier I drape a towel over the top to block most of the vent-holes and part of the open doorway. I fold towels into a soft pad and place them in a pillowcase to prevent kitten claws getting caught in the towel loops. Flip this bedding over once daily so you get two days' use out of it before washing.
*I prefer the 2x2 box as you can cut a mamma-cat access hole in the side, round enough for her to easily pass through and the bottom of the hole at least 4" above the bedding so no kittens can accidentally fall out or get dragged out by not releasing her teat. The hole opening will bump kitties off the nipples and keep them in the nest if they hold on for that long. By the time they can get through the opening on their own, they are now ready to explore their world outside the nesting box.
*another tip for using a cardboard box is to line the inside of it with shelf-paper to prevent moisture deterioration of the box.
Good luck! Piccies when you can!