If you have a friend with a lot of bits to try, that's always a good place to start. Is she nice and responsive in her current bit? Does she neck-rein? (Have her teeth been floated? No bit will work well if they haven't.)
A snaffle is always a good place to start, preferably with a French link mouthpiece, but a regular jointed bit will give you an idea if that's all you have to try.
If she likes her current bit just fine, you might try a nicer variation on the same theme: Better-shaped mouthpiece of sweet-iron metal (vs. nickel alloy), swivel shanks for a possibility of direct-reining, and better overall balance.
And I always think that an Argentine-style bit is a good starting place for a broke Western-type horse. Three-piece mouths are generally better, but again a single-jointed mouth is a fine choice too.
I almost wonder if you couldn't just ride her in a rope halter, but if you feel uncomfortable doing so, DON'T!