Your plant sounds a lot like a Peace lily (Spathiphyllum sp.). These plants are commonly used in the setup that you described; the plant's leaves grow out of the top of the container and the roots in the water. They are terrestrial plants that tolerate growing in very wet soil and in plain water; but for how long, I'm not sure. You should be able to Google care info for the plant. I grow one in a pot, so the care may be different in your case. Hope this helps.
sounds like you probabkly have two different problems.
not enought light for the plant, whatever it is.
and a betta in FAR too small of a bowl.
contrary to propaganda to the contrary, bettas do NOT do well in teeny bowls..the only people who say that is ok are the people trying to sell them.
If it is actually a peace lilly, you should take it out of the tank, plant it in a pot of potting soil, and put it in a window.
then you should buy a 10-gallon aquarium, with heater and filter, for the betta...but if you only get a 10 gallon tank, (and not larger) you cant put anything else in the tank besides that one, single betta, because its too small for anything else.
a ten-gallon tank is perfect for one betta.