Well I could have written your post last week. My LO is almost 5 weeks old. I decided to get her to sleep in her basket this week, and have managed it. It's hard though, as she did cry (still does) and I have to keep going back, cuddling her and putting her back. But, she does eventually sleep (after an hour of this!) and sleeps for 4-5 hours. Then during the night, she wakes up twice for feeds, but goes straight back to sleep in her basket after each feed. I can't get her to sleep in her basket at all during the day. She doesn't like sitting in her rocking chair, etc., she just wants to be held. So I now carry her all day long in a baby carrier. At least that way she gets her much needed naps and is less frantic in the evenings - so easy to get her into bed.
I did the same with my DD2 and it really helped. Only when I read my diary did I realise I'd done this (from 7 weeks) after weeks of hassle, so I'm trying it again.
My DP works long shifts and I hardly see him for 5 days. Also I have 3 other children, so I had to sort something out.
With my others, I found they'd only sleep on their stomach, but with this baby, she will sleep on her side/back.
I also tried swaddling, which worked for about a week, but my babies didn't like it and would struggle to free their arms and get very angry.
As long as the basket / cot is comfortable and warm, I'd say make sure your baby gets good long naps in the day and then try to get your baby into the cot or basket at 'bedtime' - even if it takes an hour each night. But she needs to be clearly tired, and you need a cup of tea to enjoy in the gaps when you leave you baby to whinge for a few mins. Honestly, it's not cruel. My father in law is a paediatrician, and he has explained how letting a baby cry now and then is absolutely not cruel or harmful at all. In fact, our baby's are in transition for weeks, so a lot of things are going to feel strange, uncomfortable, etc., for the first few or more times, and they will generally cry. But, they soon get used to them - same for sleeping in a cot or basket. Some take to it quicker than others. You have to persevere, as it's all just supporting the baby's transition from our womb into this world. It's hard, but we have to be strong, as it is actually helping them in the long term.
I also play music (only at bedtime). With this baby, I play pop/rock music. With my others I played a nursery rhymes CD. It does help I find.
We and our babies are all so different, but, I just wanted you to know I did have this problem, and don't now (after cahnging my approach and committing to it).