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The banana plant, sometimes, does not produce very much fruit during its first few years. This link will help:https://www.gardeningknowhow.com/edible/fruits/banana/getting-banana-plants-to-fruit.htm

This article will help you pinpoint the issues.

https://www.gardeningknowhow.com/edible/fruits/banana/getting-banana-plants-to-fruit.htm

If you are referring to the petals, then these will naturally fall off after while. If you are referring to the whole bloom falling off, then I would start looking at a possible phosphorus and potassium deficiency. If you fertilize regularly, then the pH could simply be thrown off. You can fix this by adding dolomitic lime to the of the soil.

Banana ID can be difficult, since there are many varieties and hybrids, but the banana enthusiasts at this site will help you:
http://www.bananas.org/f12/

IF the goal is for this to produce fruit, you will have to keep strict conditions for the plant. You will have to spray for extra humidity. If you can, you will want to keep the plant between roughly 20-26 celcius. This will ensure that it continues to grow and prepare for fruit.

As for light. This needs bright filtered light. You can use lamps with either CFL or LED, but not standard incandescent bulbs for supplemental light. They would prefer around 8 hours of bright light per day. If you wand to keep the plant alive and don't expect fruit, then you can fall outside the recommended temperatures.

Here is an article that will give you more information: https://www.gardeningknowhow.com/edible/fruits/banana/indoor-banana-plant.htm

Root hormone will not do it... Although there is some damage, it will likely recover. This season will probably be a wash, though. Here is an article that will help: https://www.gardeningknowhow.com/plant-problems/pests/animals/eliminate-gophers.htm