I really sympathise with you. My first child was a fantastic sleeper (which obviously I put down to my superior parenting techniques), followed 2 years later by the little minx who is DD who has really not been easy at all. I seriously blush when I think of the advice I "kindly" offered to all and sundry after I had only experienceed DS.
However, one of my strongest parenting beliefs - and pretty much the only one that has lasted through 2 children - is that we ALL need a good night's sleep. I know that I can't function on continual crappy broken sleep, and I'm pretty sure tht it impacts on the DCs as well. The other thing I think is that - all things considered - if your child is in a happy secure family situation, that a bit of strictness at night to help everyone acheive a good night's sleep is permissible. (This doesn't mean screaming for hours, at all, just a bit of "I mean business").
If it were me, I honestly would stop the feeding -just looking at it from the outside it seems that that is a huge root of things. But I really would consult a sleep expert to give you a tailored solution. I think also it really helps if you have an expert outsider looking in and making suggestions that either you have never thought of or you need a bit of encouragement to take.
I honestly think you should consider your own needs as well: I think you are entitled to expect a more reasonable night's sleep than you are getting and that you should take steps to come to an accomodation that you all can live with. (My view would be that your DD might want to be fed continuously during the night in much the same way that I might want Robbie Williams to sing me a lullaby each night, but it is not strictly necessary for either of us...)
My experience when we have bitten the bullet and implemented a new regime is that we have had a couple of crappy nights but that really it has taken a lot less time and been a lot less awful than we expected it to be.
If this is stricter than you want to be, then apologies. However for us the start of it getting better was us thinking that actually we all deserved some sleep, rather than thinking what mad new rigmarole we could go through with DD to make it better for her.