This sounds like our daughter. She's going to be 5 months old in a few days, but when she was about 6 weeks old, it started to be very difficult to put her back to sleep after her middle-of-the-night feeding.
It was very frustrating for awhile. I would rock and walk and she would fall back asleep, but once her little body hit the crib, her eyes would pop open and I would have to start all over again. One night, I just got so tired, I put her down between my husband and I, and wouldn't you know it, she fell asleep almost immediately.
We started co-sleeping and it was great cause EVERYONE got to sleep and a few nights after we started this, she slept through the night and has been ever since.
Nowadays, she still has to fall asleep in our bed with me laying down next to her, but I don't need to rock her, or sing, or anything, she just needs to look at me and touch my face or hold my hand while she falls asleep. After she falls asleep, I will transfer her to her crib and she usually stays down for the night. If she wakes up or fusses while I'm moving her, I just put her back in our bed. If it gets too late or if she's too fussy, she'll just stay in our bed that night.
She does not nap well during the day for the nanny, usually an hour at a time at the most and NEVER in her crib, but on weekend, when we're home, we'll do the same thing (her in our bed) and she'll nap for hours.
I'm not saying that co-sleeping is the answer, but it may be worth a try. Good luck!