It was a few (3?) years ago that I had such a full house. Bio kids were 14 and 15, one with autism. Foster kids were 8 and 9, both therapeutic level, both with serious PTSD, one with ADHD. I also had my grandchildren, aged 2 months and 2 years when they came to me. BigGirl was the 15 year old, and absolutely amazing with the little ones - I really didn't ever do this alone.
So here are some of the specific things we did:
A large list on the wall of things that are always OK to do when you have free time (read a book, color, eat a piece of fruit, clean your room, work on a jigsaw puzzle that is already out, etc), things that are probably OK, but you have to ask (eat something else, set up Legos or toys in the living room, watch TV or play a video game, play in the yard, etc). Each part of the list had dozens of suggestions - we made the lists together, and added more frequently. If someone was doing something not on the list (annoying a sibling, making a mess) I could say, "Go find something on the list to do". Sometimes I added things to the list as a surprise : "Look in the freezer" when I had bought popsicles.
But really, they didn't have much free time. The foster kids had therapy and family visits each week, and were on the soccer team. I often drafted someone to help with dinner or set the table. We didn't have set chores, but kids were generally pretty agreeable about helping when asked. Since the purpose of asking for help was really as a distraction more than to get the job done, I tried to make it fun and cheerful, not a chore or drudgery. My kids got into major trouble when left to their own devices. Beyond the normal, I am talking fire starting and worse. So they were never unsupervised at all - really 24/7, line of sight. Yes, it was exhausting! It did get to be habit to keep everyone occupied all the time. That seemed to make the biggest difference. We made many, many park, museum, zoo, and library trips when I couldn't think of anything else to do at home.
We had posted schedules - a big white board calendar, with each kid's activities in a different color pen. Morning routine, down to the details (get up, use the toilet, get dressed, have breakfast, brush teeth, gather coat and backpack, go out to the school bus) in 5-10 minute increments. Same with the evening routine, from dinner till bedtime. Schedules for each kid were different, so even bath turns were planned and written down. The 3 high needs kids, and the toddler, seemed to thrive with this predictability.
I hope some of these suggestions fit your family. Hang in there - it really does get better!