Poor you!! Went through this with both of my two and remember how dreadful it was.
Take up your friend's offer. It may seem like a big sacrifice to you, but any friend worth their salt would do it. When you're in a state, things like that seem like big favours but they aren't really. I remember refusing my mother's offer to look after our screaming colicky dd one night so I could sleep. Weirdly I refused-felt that she wouldn't be able to cope.
You WILL sleep eventually, when you need to.
It's always said I know, but you need to forget about routines and day and night for a while. Sleep when the baby sleeps. if you lie in bed all day and are up all night, what does it matter?
Book a massage/reflexogy treatment. I couldn't sleep when pregnant and was desperate. my sister massaged my feet for me-worked like magic.
Try a stiff drink. A hot toddy worked for me a couple of times-whiskey, honey and lemon. felt all woozy and relaxed afterwards. Don't overdo it though (obviously!!)
Never drink spirits but this worked.
Keep chatting on MN. It will keep it all in perspective.
Hang in there-it will get better. XXXX