How many watts is the light source? If it's a vivid green leaf, it probably needs a LOT of light (read: more than a basic tank setup has).
You can try using floral fluorescent bulbs, they're made for plants. The fish don't "pop" as much but plants love it. Makes a huge difference. but be careful cause it does a huge difference for algae also. I've stopped using those bulbs for that specific reason.
Plants, when they don't have enough light/CO2 will last like 2-3 weeks so if it's dying, it's probably not a good plant for your setup.
What kind of gravel are you using with this betta btw? What colour is it?