I was going to try the same thing but instead elected to go outdoors with my setup. There are meters that can measure the light shining into the room but I would just give it a try. The system is small enough that you can move it if you run into a problem. Don't count on the fish tank light as the color temp of the light is not known and it is probably too far away and indirect to help much.

There are a few little issues that need to be addressed if you want to grow with exposure to the sun.

1) If your reservoir is exposed to light, it can't heat up, otherwise tall the dissolved oxygen will leave the water and anaerobic bacteria, which will kill your plants.

2) Algae spores are floating through all our air and will land in your system. The problem with algae is that they need light nutrients and oxygen. So, they will take all your dissolved oxygen, causing root rot again. They will also clog your pumps and cake all your pots/trays/res's (just a pain to clean). Algae grow where theres light, so just make sure the water and trays aren't exposed to the sun.

Last year I had a floating hydroponics system to grow lettuce outside. A couple of items to consider. You will definetly want an air stone in the bottom of your tank attached to a aquarium air pump to provide sufficient oxygen. Also, make sure no light can get to the solution, this will minimize the growth of algae/bacteria/fungi. In addition, ideally you will want a good nutrient solution, and you will want to monitor the pH and TDS (total disolved solids). The nutrient solution will need to be changed regularly - every 1 to 2 weeks.
I agree with a previous post, that you can try this inside if you have good outdoor lighting. Fluorescent/fish lamps will not be sufficient to grow healthy plants.

If you want to grow mature houseplants in hydroponics, the lighting requirements are no different than if you wanted to grow houseplants in soil. Algae is a problem almost all hydroponic systems face regardless of what light source you use.

Most houseplants do not grow very much because the limited lighting conditions are not condusive to vigorous growth. Because of this, houseplants need a much lower nutrient concentration than more traditional hydro crops like toms and lettuce.

I disagree with what indywar said about the solution needing to be changed so frequently. If you are regularly monitoring and adjusting the pH and TDS, there is no reason to change the nutrient solution every two weeks.

My sugestion is going to your local library and reading everyhting you can find about plant physiology. An understanding of how a plant's roots absorb oxygen and nutrients, and how a plant's leaves use photosyntheis is key to long term success as a hydroponic grower. So many people grow plants in hydro but don't fully understand how it works. Educating yourself about the physiological workings of plants is the single best thing you can do to advance your hydroponic knowledge.

Hey sdrawkcab happy early birthday!
Same day as mine.

Thanks. :)