Yep, was going to say that laughing is a really good way to get their mouths open.
And I second malamamama's applause -- really great that you're making such an effort. It's important.
I think my dd was just a little bit older when we started using this song:
We brush brush brush our teeth
We brush them left to right
We brush them up and down
To get them clean and white!
At some point, ya just gotta do it. A frequent analogy here, frequent because it's a good one, is changing diapers/ wiping poopy butts. Most kids go through a phase or twenty where they just cannot stand to have their diapers changed/ butts cleaned, but it's not really an option, ya know? If they aren't cleaned, bad stuff happens. The same with teeth. A little less linear -- sometimes people are really cavalier about tooth cleaning and the kids' teeth still turn out fine -- but if you have the risk factors, and sounds like your daughter might (good for you for figuring that out), you just gotta do what you gotta do, as hard as it may be.
My dd was older when we started really brushing her teeth -- around 19 months (after we'd found 7 cavities) but once she figured out that it was just going to happen, she really calmed down about it. Hasn't been a big deal since.
Good luck! So nice to see a question about prevention.