Cindi - sleeping through actually only means 5 hours! in medical terms. So don't panic. I would say that my baby was sleeping through when he went to bed at 7, woke at 11pm for food, straight back to sleep, awake again at 2am for more food, then brought into my bed.
I'm very lucky that he now sleeps 7pm-7am (although he is teething at the moment, so it's gone a little pear-shaped!).
At five weeks, I wouldn't worry too much. I think he needs to go to bed a little earlier. Really, he should be in bed anywhere between 6pm-8pm. If he's in bed by about 7ish, feed him at 10.30pm and then again at about 2-3am
Are you bottle-feeding at night Cindi? - if you are, try to feed him before he wakes at 2am, this is called a 'dream-feed' - have a look on the internet for more about dreamfeeds. If you are dream-feeding, you actually shouldn't need to burp baby.
Kelsey - I did controlled crying with my 5 month old, but I really think that leaving a 10 week old to cry (and by cry, I mean wailing and screaming, I think it's ok to leave them to 'fuss' a bit, but no longer than 5 minutes ) on a regular basis is not a good idea.
To be honest, I would also leave out the story. She is a little young, and this is 'stretching out' the bedtime routine quite a lot. You might find that if she was in bed earlier, she'd sleep better.