I have a 5 gal hex tank with one betta in it. I have one java fern
that has survived a couple of years, but can hardly be considered to
flourish. I have tried amazon swords a few times, that seem to slowly
die.

The tank flourescent light is on from around 8AM to 10 PM everyday,
which I guess is considered a lot of light. Water temp is around 78F
in winter and 84F in summer. Nitrate levels are consistently around 10
ppm. Tank has a small biowheel filter.

Any suggestions on a plant that will flourish in these conditions or
what should I be doing different?

Thanks,

Todd K.