I have a few suggestions, but I'm not a professional therapist, it's just observations based on the thread.
First, you are being very very harsh in expecting yourself to feel gushing love for a child after being up all night with a vomiting child/parent/waking every hour. I think if you were to get the sleep sorted, then you might get enough energy back to start to deal with the emotional side and I would go for sleep training as a solution to this. I did it when my child was 7 months old and I crashed the car. You can't function on poor sleep and she's a shit sleeper in your words. This coloured my life when I was exhausted.
Second, you may well be depressed, I would say that feeling of flat numbness, dragging yourself through the day, going through the motions, is a very good description of depression, you don't have to be angry, flat is often how it comes out. So, definitely go to the GP and start talking- either talking therapies or anti-depressants or both may give you the much needed lift after a very difficult pregnancy and PND period.
Third- although there is some truth to the 'fake it til you make it' I think you can take it too far. Once you start living a life completely opposite to your true self, you start to feel more and more distanced and weird. I'm not saying don't cuddle or snuggle or have fun feeding. But if you are relentlessly trying to do this all day every day, you are setting yourself up for failure. I believe strongly in authenticity in parenting, so that it's ok to be yourself, and not feel always like doing stuff with your children. I think if you are too self-sacrificing and always just put your own emotions and feelings and poor health to one side, then that resentment bubbles up- or you just suppress and go flat, and I wonder if that has happened here. I would not try to be constantly interacting, clowning about, cuddling and having fun if actually you feel dead inside. Do a bit of it and then relax and sit back. Your baby can still crawl on you or hang out together. I think you are probably exhausting yourself pretending to be a great mum when you feel awful, and allowing yourself a window to be not great, with a few moments of greatness would be just fine too.
If you then did some play therapy, or even just set aside time to have fun together, but there was a timed end to it, it wouldn't feel like this endless period. I find playing quite boring anyway and didn't do that much on the ground pretend play - so I did lots of things I do like, like going to the park, reading or watching TV together, and left my husband do to the more active/imaginary/fun days out stuff. You don't have to do it all yourself, and being yourself is also important too, even in the mothering relationship.
I think this is fixable if you tackle it from many fronts, and I'd start with the sleeping!