Ok, this is one reason why we only had 1 child.. I have to be able to sleep to function.
I know you are exhausted and very frustrated.
These are the things we did to help our daughter sleep through the night. It worked at 6 weeks..
Sleep with her crib sheet one night or at nap time. Place this on her crib mattress.
At night, give your baby a nice warm bath. use a low light in the bathroom.
Use good long strong strokes on her arms, legs, head.. Like a massage. No horse playing. Make it as calm as possible. Take her to her room, with low lights, maybe some soothing music or white noise.. like a fan.
Try not to talk to her. You could put some lotion on her again with nice long strokes. Place her in her pajamas.
Then feed her a nice sized bottle. No eye contact look away or close your eyes. No talking.. let the music play. Make sure you get every burp out of her.. In between burps, rub her back in upward motions.
When you place her in her crib, lay her on her tummy, with her head up against the corner of the crib, head against the bumper.
Make sure there is no light in the room.
At night when she wakes, let her cry for a while. If she does not calm down, very quietly and is as dark as possible, go in and pat her bottom to kind of rock her forward and backwards.
If she needs a diaper change, do it in the dark, do not make eye contact do not talk.
Maybe place a night light in her room or leave her door open and only allow the light from the hallway to come into her room (which ever one is the least light you can work with).
If she needs a lovey, look at what you have that is soft and not too thick. Sleep with it one night or at nap time. Let her play with the lovey during the day, place it in her stroller or in the car seat with her. This can be a blanket, a stuffed animal, some children like something with a silky texture on it.
You need to take away anything that gives her stimulus. You want this time to be so boring she goes right to sleep. Soon she will be in a pattern of longer sleep.
I am sending a good nights sleep.