Offer to do things, but be specific, not just "whatever" you need. Laundry is huge with that many kids, take it to your house and do it. Try taking just one or two kids to do something and arrange someone else to take others. Individual attention in a large family is always appreciated. If you are baking cookies, invite one or two of their kids to help, rotate who gets to do something.
I agree in helping the family do for themselves but dad might not be up to that right now.
With mom depressed don't be surprised if the house is a wreck, try and hide that your reaction. The kids can help get it in order and learn to help, yes make it a game.
Meals are such a blessing, approach dad and ask the families likes and dislikes, ask specifics: do you eat fried food, vegetables, pork, chicken. I have found many say we eat everything but when I engage them in conversation they share we don't like..........
Visit mom and let her know others are pitching in to help with the kids, that will ease her concerns.
God bless you for being such a good neighbor.