I recently purchased a java fern, moss ball, and banana plant. I also have one fake plant. After doing some reading, I realized that my lights that came with my tank may not be strong enough for the banana plant. I was thinking of adding this to my tank http://m.ebay.com/itm/270975018457?nav=SEARCH&sbk=1
Would that be better for the banana plant or is there a better type of lighting I can add to my tank? It's a Hawkeye 5 gallon aquarium kit.

It might help a little, although those LED flexible strips are notorious for dimming or entirely dying after a while, particular when installed around water.

I'd give it a while and see how it does before you worry about it, or just pull out the banana plant and stick with low light plants like your Java fern or some Anubias.

I agree to let it be and see how it does. I have banana plants in my low light tanks and they did great after a while. Gotta give it time to adjust.

The banana plant is shooting out two new leaves. I hope that's a good sign!