I don't see why a nanny can't throw in one load of clothes per day (even if it's just the kids' clothes), and then involve the 6 and 9 year olds in the folding. Great skills for kids.
I don't see why she can't take something out of the freezer to defrost if you forget, or put on a big pot of water to get it started for the pasta.
She and the kids can pick up toys. Everyone can have one bin in the family room or playroom, and they can put their own stuff in there. Put a photo of each child on the outside of each bin. We also put a photo that helped with sorting - cars/trucks in one, Legos in another, general blocks and building toys in another, animals/dolls in another. If your kids share toys, that works fine, and it bypasses reading skills. Get some vertical dividers from the office store that help kids shelve books without them falling over.
I'm sorry your husband is overworked, but it sounds like you both are. Why do cooking, cleaning, and straightening up all fall to you?
Set aside one day on the weekend for family cooking/chopping/prepping, and let everyone help. Pack up some packets for the crockpot (if you don't have one, get one) and throw things in before you go to work or have the nanny do it. It's pretty easy to say, "Take out ziploc bag #2 and add 2 cups of water, set on low for 8 hours." Take 1 hour on the weekend for cleaning - you and Kid 1 take the top floor, hubby and Kid 2 take the bottom floor. Switch it up every week so each kid works with each parent, so everyone works on each floor. Do major cleaning once a month, but bathrooms and sheets and towels get done once a week. Put a laundry basket in each kid's closet. Put individual color-coded laundry baskets in the laundry room and let them all fold/sort and put away their own things. By age 12, they start doing their own washing, maybe sooner. The younger one does the trash and recycling so all the work doesn't fall to the older one. On grocery day, it's All Hands On Deck - everyone brings stuff in from the car, unloads the bags, and starts the sorting according to the way your kitchen and pantry is organized. For us, pasta and soups go in one area, canned goods go somewhere else, fruit goes on the counter for washing, and of course refrigerated stuff gets unloaded first.
And relax your standards. It's okay to close a door to a bedroom, it's okay not to make beds every day.