I feel your pain. Huge hugs to you. Its so bloody hard when they just won't let go.

My DD2 was very much like how you have described your LO. Occasionally (and sorry, but only occasionally), I would give her a feed while propped up on pillows on my bed. She would then doze off on my shoulder or chest and if I was very, very careful I could slowly slide down and eventually roll her off me onto the mattress. The bliss never lasted more than 30 minutes at the absolute most, but it at least gave me the chance to have a very quick shower, make a sandwich, or god forbid just be limpet-free for a small window of time. If your LO prefers being across your lap can you maybe put a blanket, flat nappy or something across your lap with her on top so you can try and drag her off all in one when she's asleep?

Big to you. New babies are exhausting enough without the added nightmare of one who you can't put down.