Try lowering the noise level during the day. I know that sounds wierd. but if you turn on the vaccuum cleaner, it's noise overload, and a baby will check out -- go right to sleep.
Start on a weekend, and enlist your children to take turns and play with their sibling while you get some rest. Dad can join in, with the baby on his chest while he watches sports on TV. He can also take over baby responsibilities the minute he gets home from work, and you can make supper, nurse the baby, and go to bed. It's good for him to have daddy-bonding time, too, and if you get some sleep, you'll have a better relationship with him, as well as with your twins and infant. Ditch the chores. Who is going to remember 10 years from now that you skipped doing chores when you were too tired to stay awake ? Your health is much more important to your family than a carefully crafted meal, or a perfectly tended building. Home is family life, not better homes and gardens.
I don't do the "cry it out" thing, either -- esp at this age. I did, however, prefer to sleep with my child between my husband and me over not sleeping at all. We slept in a queen sized bed, and there was plenty of room for a wee infant. As our babies got older, they went to bed in the crib, but because I worked full-time, I was too tired to stay awake while they nursed at night, so I cuddled them in bed with me, and they nursed, and I half-dozed. We would usually get about 5 hours of good sleep without the baby, then a few in the early morning when we were conscious of having company in bed. If you don't want to do that, you could go to bed with your baby earlier in the evening, fall asleep, and have your husband move the baby to her crib when he climbs in. I think it is more important to get the sleep YOU need to cope with your child, and your family, than it is to worry about whether or not your infant sleeps in your bed temporarily.
I would not do the "no eye contact" thing at feeding time. All you are doing with that is frustrating both of you when you should be building your relationship. (and obviously, you don't enjoy that at all.)
You could try heating the crib sheets with a heating pad, and fully removing it from the crib before you put your baby down. The warmth might help her to stay asleep. also, if you cuddle her in a blanket before you put her down, keeping her in the blanket, in basically the same position might help. . . . . and then there's my sister's method. she would hold her babies until they were so well asleep that she could pick up an arm, drop it and have it plop down, totally. No, half hearted, slowly lowering the hand that implies some level of awareness and muscle movement. she would wait until the kid was so far gone that the hand simply "fell" to the baby's side. At that point, she knew the baby was so well asleep that putting him or her in the crib would make no difference at all to the baby.
Bsbies seem to quite naturally be up all night in the womb and sleep during the day. When they find daylight, it doesn't necessarily change their habits. I always found that their habits changed when my habits changed -- like when I returned to work FT and they had no choice but to do what I was going to do, because I couldn't alter my schedule to theirs. My kids generally landed into a schedule of a morning nap, an afternoon nap, and bedtime around 10:00, because that's when I went to bed. If they laid awake while I slept, I didn't really care. If I nursed them longer to get them to sleep, I didn't really care. But I did go to sleep. I had no choice if I wanted to keep my job.
the twins will be home from school for vacation soon, and they can help you. Ask them to help play with and entertain their sister, esp while you "do chores", and use the family time as a time when everyone bonds as well as helps you cope with baby. As the little one becomes more and more able to digest stimuli, she will find that being awake is fun, during the day, when there are more stimuli to enjoy. This time period you're in right now, is horribly exhausting, but it isn't permanent. In another month, you'll begin to see a better sleeping pattern emerging, and you'll begin to feel better.
Meanwhile, Merry Chtistmas ! Won't you have fun with the 8 yr olds, and won't they have fun playing with the baby's toys ?! :-)