You CAN do it.
You need to lower your standards. Right down.
When I was having no sleep after the arrival of DS2 my target for the day was "everyone is fed, nobody is dead". That's the kind of target you can achieve.
If that means the TV on all day for longer than you normally would, so be it. If it means you just put coats over their PJs to get to shops, so be it. If it means one or both children has to be strapped into a pram crying for half an hour so you can do what has to be done, so be it. If it means toast for three two meals a day, so be it.
You do not have to be a brilliant, engaged, stimulating wonderful mother for every day of your children's lives. A few crappy months weeks after a new baby arrives and everyone is knackered is NORMAL and not going to do any lasting harm.
Make sure YOU eat, go to the loo, drink plenty of water etc as well as sorting them out. Make sure your partner knows how important it is for you to know when they will be getting home and that you NEED them to be on time.
This is the toughest stage and it does not last forever, I promise. Don't beat yourself up too much. In a few short months the baby you can't put down and the toddler you can't unpeel from your leg will be crawling after each other round the living room pretending to be lions and you will suddenly think "Aha! THAT'S why I had another one!" and it will all be okay 