Ok - here's my suggestion. You may want to make sure you're sitting down.
Put her on her stomach. Make sure there are no stuffed animals, quilts and such, and just put her on her tummy. I know this freaks people out. But, as I seem to be a little older than most of the new moms out there, I can remember when the rule was NOT to put babies on their backs.
If this makes you nervous, try it at a nap time when you can go back and check on her frequently. My oldest (now 6) was a terrible sleeper until I decided that I couldn't stand it anymore, and put her on her tummy. She went to sleep quickly and slept better and longer. My husband FREAKED!! He's older than I am, but he's a youngest child and I'm an oldest - so I've seen this in action - guess he hadn't.
When my second (now 17 months) was born, I put her on her tummy pretty much from day one.
Babies are curled over themselves when they are inutero. If they are on their tummies, they can recreate that feeling - and that is a warm safe feeling for them.
Those are my thoughts. If people want to send me nasty grams, so be it. But, my advice has worked for hundreds of years! It's been about 25 years that the back sleeping thing came into play.
Whatever you (and you H.) try, I wish you luck. It's SO frustrating to hear them cry when you know they are tired and sleep would make them feel better!