I feel for you. When Lilith had her kittens 2 yrs ago, it was right in the middle of our bed. I was in the bed, asleep, when it started, and BF was in the computer room playing a game. I woke up to a teeny tiny "mew!" and turned on the light, well there was Jules! I got up and dressed in a hurry, and over the next 5 hours there were 5 more kittens! We didn't get much sleep that night, and everything down to the mattress pad had to get thrown away, as it was soaked in birth-stuff.
Once she and the kittens were settled, we were able to get them moved into a large crate we'd set up for them in the closet (sounds like the situation you've got going there). Everything went fine.
Hope your cat and her babies are doing well! Let us know, and we want pictures!!
What color are the kittens? What color is the momcat?