Big ((hugs)) for you.
It must be really hard work with two other children to care for at the same time. I really feel for you. I remember spending a long time getting my LO to sleep and he is my only one, I truly don't know how anyone manages with more!
It's true she is young, and it's going to take some time yet before she has learned to sleep well.
I had my LO in a travel cot in the living room for the first 10 weeks (at which point he refused to sleep and I tried taking him up to his bed and he went off brilliantly, so he had chosen to move to the quiet upstairs). This meant that I could more easily 'continue' with other things and also be right there if he needed another shush or pat. Sure, I didn't have two LOs making noise, but the house was not kept silent for his naps. The TV or music would be playing and I sometimes had my mum over who is not the quietest of people. I'd also get on with things like vacuuming or cooking whilst he napped.
Do you think something like this might be possible? A travel cot/pack and play or a pram somewhere close to the living area to enable you to put her down (you could rock her in the pram) but still be available for your other two?
The sling is great, mine took naps out and about in the sling and once or twice at home when I was having difficulty settling him, so at least my arms were free, but I know even small babies get heavy!
I also remember it taking the full 20 mins (as Tracy said) for him to fall asleep, I'd swaddle, cuddle and calm then pop him in his cot and I could have set a clock by it...I often sang as the shushing I found hard to keep up for 20 full mins (but some people record it and play back, or replace it with white or pink noise, sounds much easier!), but at the end of the 20 mins his eyes closed. I can also remember the more difficult times of it taking an hour. I would sometimes rock the travel cot a little if he started rousing, which I felt was preferable to picking him up because it maintained the feeling of his own weight on the mattress and I could gradually wean. Sometimes I also found he would wake up after a short time but didn't cry, he just lay there, swaddled, gazing, and would continue to gaze for ages, I think he maybe felt comforted by the household noises and knowing I was there, he sometimes fell back to sleep after a long time gazing.
Not a perfect solution for you, just thoughts. Like I said, at 10 weeks he chose to move to his night time bed so I personally didn't have a problem with the transition. We had an amby hammock (for his night sleep) which he found very comfortable and when he woke in there his own body movement rocked it slightly which helped to lull him back off to sleep without my help. You cannot use a swaddle with the hammock (for safety) so he was swaddle weaned at 10 weeks too.
Maybe others have some helpful tips too.
Have you looked on the twos board I think there are some helpful tips there
http://babywhispererforums.com/index.php?board=20.0