I don't think it's fair.
I have family who manage to combine utter slobbishness with pretty happy and well-adjusted kids. The house is a WRECK. But the kids are well fed and reasonably sane.
It's not ideal, but it's not to the level of "kids need to be taken away," not by a longshot.
I wonder if there are any church or nonprofit groups that might help you.
Other than that? Toss out any clutter that makes it less easy to clean.
The kids can help, probably need to be trained to, as there's five of them and one of you.
BUT you need to train them using *only positive reinforcement.*
I think this would work?
Basically, if they do a bad job, you praise their hard work...and go back and fix what they didn't get.
If they actually do it RIGHT, even more positive reinforcement.
Had my mom done this she could have slowly and pleasantly trained me into doing almost all of her housework, just the way she wanted it, by the time I was 9 or 10. I just wanted her approval.
(As it was she was really looking for excuses to take her crappy feelings out on us. When one of my alters figured that out, we collectively just egged her on until she hit us and screamed. Once we got it out of the way for that day we could then go to our room and do really warped stuff with Barbies or whatnot. We were disturbing kid(s).)
You just CANNOT get angry at them for doing chores slowly and badly...like a little kid. You CAN gradually raise the expectations you have for giving the EVEN MORE positive reinforcement, as the child acquires skill (A kid should always feel loved no matter what).
If they get resentful, ask them why, because that actually may be a problem. I don't think it will work if they resent it.
Not saying its' 100% a guarantee, but little kids will do lots of stuff just for approval.
Older kids...dunno. At some point the ship has sailed. Might be easier to just pay them by chore, if you can.