I totally feel for you and your wife! My husband and I had our second child this past January, and we were not prepared for how different he is from our daughter in regard to sleep!
My husband would stay up all night with him, and sleep during the day b/c of it. Thank the sweet Lord he was not working at the time! I needed my sleep so I would be able to attend to both my son and then 3 1/2 yr old daughter.
Don't be afraid to pull out the baby swing during the evening. We had Benjamin's swing in the middle of our tiny living room for the first two months. It was the only way he could sleep at the time. We didn't even transition him to the crib until he was about 4 months.
I think the white noise is a great thing you're doing. It is what your little girl is used to from being in the womb.
Also, gardenandcats is correct about newborns preferring their sides to sleep. We also rolled up small towels and positioned them against our son's back so he could be on his side. This worked very well.
And of course, don't forget the obvious - Lily is going to wake frequently because she doesn't yet know how to put herself back to sleep and b/c your wife is breastfeeding. If you know she is not hungry and just waking, you can try a pacifier and checking the temperature.
My kids always do/did best when they're in footy pajamas and no blanket. We keep the temperature at about 68 degrees, which is comfortable to us.
You're probably tired of my advice, but I just want you to know that it's not just you!
It's your right of passage that is SO short lived. And remember you cannot spoil a newborn.