Starting to struggle with coping

Baby is 10 days old, I'm so in love with her. Everything about her. I feel like she's the missing piece that I've been missing but I'm struggling with feeling very alone. My partner is off on paternity leave and he's been trying to fix up outside the house, instead of bonding with our daughter. During the day he barely goes near her. At night she has to sleep on me or she won't sleep at all. If I need to nap during the day i have to bring her with me. If I have to go to the toilet, she is with me. She's been quite sick so she's been quite hard work the last few days and I'm suffering. He hasn't changed her once without my help. And hasn't attempted it once in the past 3/4 days. I'm having to ask just for him to hold her for 5 mins just so I can brush my teeth and wash my face. I had an emergency c section and the recovery for that has been shit for me. I had a traumatic birth and was in hospital 6 days before she was here and 5 days after. I'm running on zero sleep pretty much, I've had absolutely no time for myself. I feel like I'm suffocating already. His excuse is she's too little and fragile. Too scared to hold her or change her once he hurts her. I've given up trying to make him do anything. This evening I could feel myself about to combust with exhaustion and asked him just to watch her for 30 mins while I could try and sleep. She was sound asleep on my chest and I tried to put her onto him and he just told me to put her down. Obviously she woke up and screamed.... obviously he couldn't deal with it... obviously mum to the rescue again. To be honest we don't have a great support network, my family are very toxic and I don't really want them heavily involved in my daughters life and his live 2 hours away. So it kinda is just me. I'm so burnt out now already, it's gone past the point of catching up. I just don't know what to do. I don't even know what the point of this post is.