One reason they need a mixed diet other than high-protein foods like bloodworms is for the roughage found in stuff like flakes and pellets. The key word is FIBER, which keeps them (and, in fact, a great many species on the planet) from getting constipated. It keeps things moving, so to speak.
Since few Bettas seem to like vegetables (mine are always much less than impressed at the idea), we give them pellets or flakes. (or frozen Peas!)
There are many ways to convince Bettas to eat new foods. One method is peer pressure. You want one Betta to eat something that another Betta already eats? Have a divided tank handy, maybe? But the picky Betta and the other Betta in the divided tank, one on each side to prevent aggression. Then feed them the food item you desire them to eat. It may take the picky Betta a little while to catch on, but usually, the sight of another fish eating will make the Betta realize that there is food present, and that the stuff that fish didn't think was food actually is.
A more drastic way is to wait them out. This is usually best used when the Betta won't eat anything you give them, such as when they have just come home from the pet store and the stress has them forgetting they have a biological need for food. Fish are cold-blooded, making their need for constant caloric intake much less than that of us mammals. They can get by without food a lot longer (proportionately) than we can. So here's what you do. You offer the food you want the fish to eat 2-3 times or more a day. If the fish does not eat the food, remove it so that it will not foul up the water. Try again later. If the fish does not eat it again, simply remove it. Sometimes this goes on 2 or so feedings. Sometimes this goes on as long as 2 weeks. The fish will usually crack and eagerly devour whatever you're giving it (as long as it's an actual food), after a good inspection of it. Of course, if the fish DOESN'T crack, don't let it starve to death. Give it a while of pity feedings of whatever you can get it to eat, and then you can try the cycle again.
You can also cave in and buy live or frozen foods such as brine shrimp, bloodworms, blackworms, daphnia, etc. Most Bettas, no matter how picky, will accept live or frozen foods. Some even eat crickets. Many of mine do. This fact was learned by accidentally dropping a cricket in the fish tank while trying to feed the lizards.