Imawurzel, I have to disagree with Smamfa here. I see where she's coming from, and it's fine to acknowledge that they need to learn to do it on their own, but all babies learn this eventually without being taught, and it could happen for you tomorrow! Cry-it-out or controlled crying are not recommended in young babies because of concerns about potential damage to their emotional development. It used to be advised that babies were no less than 6 months of age but I think now most authorities on the subject actually say 12 months.
For what it's worth my DS was exactly as your baby is - totally relentless and hell without sleep, but hellish to get him to sleep. Around 20-22 weeks he became better at being rocken then laid down to sleep (before he was held most of the time because it was the only way he'd sleep). Nothing we did changed - he just got better at sleeping more deeply. He was really windy too and I can suggest oodles of things that might help sore tummies if you like!
My advice (well, it helped DS over time, might help you too) is to incorporate patting into the sleep thing consistently. This way you can work on putting her down while really really sleepy but the familiar pat helps her stay asleep as you lay her down. Then you can gradually put her down as she's less and less sleepy and pat her off to sleep. This whole idea was laughable when DS was little but from 5/6 months it became useful and now we use it daily (actually am now trying to wean the patting but it sure beats rocking him in a sling for an hour at a time!).
I started by BFing lying down but patting his bum as he nodded off. Then I patted as I rocked. So pat = sleep. Then I would rock until drowsy/eyes closing/floppy depending on good day/bad day. Then put down and keep patting until kaput and run away! Now I feed a bit, rock a bit (just for a minute on good days), lay down and pat for a few minutes, then sit by the cot for a few minutes, then run. it's definitely gradually getting easier. A long way from self-settling but I can be back downstairs in 10 mins on a good day and he doesn't wake quite as frequently so it's helping with night wakings.
I know how utterly soul-destroying it is, I honestly do, but 20 weeks is still really, really tiny (you'll realise how tiny as time passes, but it's hard to grasp when you're at that stage, I remember feeling like DS should be able to do so much more than he was doing back then). You will get there, I promise!