I"m sorry you're having a rough time, it definately can be tough dealing with illness and having to tend to the house and kids.  I can relate, I have fibro, Lupus, arthritis and a mirage of other things wrong with me - so I totally understand where you're coming from.  
Best I can tell you is to do what I do - make a list and do a few things on that list each day; assign a few things each day.  
Dishes, well, those can be done daily, yes - but how about paper plates once or twice a week to help alleviate dishes?  *not the most environmentally friendly I know, but...works for us*.  
Cleaning up around the house with kids:  Pick a song that can be your "clean up song"  We used to use Barney when the kids were little "Clean up clean up everybody everywhere.. clean up clean up  everybody do your share! 
"  and we'd take the entire song for 2 cycles and clean up all the toys - both before nap time and then before the end of the day.  Worked wonderfully.   The older kids can help out quite a bit actually - they are 6 so they can help take trash out and set up for lunch/dinner/breakfast.  No reason not to.  
Quiet time:   HAVE this each day for ALL the kids!   for at least an hour - you can put a "quiet time movie" on and have all the kids sit quietly and watch it.  If the youngers ones take a nap, then you sit the older ones down for "quiet time" during this time.  but..either way.. you must make this hour of quiet time every day.  Its when YOU can get some of your things done without having kids pulling at you.
If your own children are the ages of 4-6 years, they can help feed the animals, gather eggs, etc.  It might not be as fast or neat as you'd like it to be, but it can be done.  They can learn to use a swiffer duster and help dust 
  Kids LOVE using those things - trust me.  Use things that work for their ages - how about a chore chart with a reward system like whomever gets 5 golden stars gets a special toy out of the box?  (you can get a bag of something from the dollar store; candy, small toys (age appropriate) or even fun special alone time with Mom). 
Bickering:  Have a time-out mat or stool - have a few actually - that way you can separte the kids.  Use a time out equal to the age of the child (2 minutes for a 2 year old...6 minutes for a 6 year old, etc).   It works... might take you a while and them a while to learn, but it does work.   
As far as DH goes - well - if he's done the dishes 3 times in 3 days - thats pretty darn good all things considered!  At least he's helping.  Praise him for helping instead of bashing him for not doing enough - remember he's working too and pulling long hours, just like you are.  Is he with kids all day - no - but... he IS working.
Just remember - life is too short to be bickering about house work.  Pick a few things to get done each day and stick with those things only.  So today you vacuum and sweep - tomorrow you dust  and may sort the laundry.  The next day you do 1 load of wash or maybe 2.   You have to pick and choose what you do here, being sick and having kids, its the only way I get things done.  And there are some days I do absolutely NOTHING around the house, the dirt will sit on the floor and I'll get to it tomorrow.  Why? because I hurt, plain and simple.   The laundry might not get done on my laundry days of Monday and Weds - so I may do it on Tuesday and Friday instead.... but... eventually it will get done.  Does DH vetch to me?  NEVER.... because he knows better, if he vetches at all, he'd better be helping me out (and he does - I'm very fortunate) but it wasnt always that way, believe me.  
You'll get  through it honey - just put a list up and do what you can.  Just remember - Rome wasnt built in a day... and this too shall pass
Lots of hugs and encouragement to you!