My baby won’t sleep without me at night.

All day she naps by herself in her pram or basket, but at night she will literally stay up all night unless she sleeps on/near me. When she’s fell asleep I’ll lift her to her bed and she will wake up within 5 mins, start wining and won’t stop until I pick her up. She’s only 5 weeks and I don’t want to risk co sleeping im only getting 2-3 hours sleep in a 24hr period, and my fiancé is sleeping anywhere from 10-14hrs and won’t even offer to do one night, he gets 3 days off a week, I don’t expect him to get up with her when he’s at work as he starts work at 6am but on his days off I’d like a little help but he doesn’t even care to ask and if I ask him to he will say yes and then fall asleep as soon as we are in bed so he doesn’t have to. I don’t know what to do, my mum had her today for a couple of hours so I could sleep but i ended up catching up on housework. And I feel so bad because I’m just so tired I can’t be bothered to look after her I do, still do everything she needs because I have to and she needs me. I just don’t know how to get him to help me. I’m exhausted and then when I sleep in the day he moans that nothing gets done. I can’t win.