Another view

Do you think you could post a better photo and one that is oriented correctly? That will make it easier to offer suggestions. It sounds like an awesome idea though!

You might want to choose between green leafy plants and fruiting plants like tomatoes, and just grow one. They require different temps, light spectrums, and light intensity to do well. It would be easiest to start with lettuce, herbs, and greens.

If you can afford the price, the best lights right now in terms of quality are digital ballast hid lighting. You'd want metal halide for leafy growth. I would just hang up one big light that is over your head and out of the way. You have to be careful of the heat build-up. You might have to make an exhaust vent to blow out hot air.

Spiral florescents are my next pick for lights, but you need several of them and they need to be very close to the plants.

I think if you had some small shelves that are about as tall as the window sill you could grow plenty. If the shelf ended about 10 inches from the sill the plant would be in the sunshine most of the bright part of the day (assuming its planted in a 10 inch tall pot).

You would want lights overhead of the plants during the evening when winter days are the shortest. Plenty of plants will survive without the lights but they will barely grow and probably won't bloom and fruit during the shorter days. You could use regular adjustable floor lamps with the bulbs set within a foot of the top of the plants (the closer the better) and either use grow bulbs or fluorescents. Lighting is tricky. People seriously growing under lights use big bulky expensive lamps that won't work well in a home-ey sunroom if you want to also use it as a reading/sitting room. I would at least start out with something simple and expand from there as your experience grows.

You'll probably have better luck growing a cherry tomato in a pot outdoors in the summer and then bring it indoors for the winter and toss it out when it goes down hill - it may last all season but they usually get disease or insects before March. I can never get regular tomatoes to ripen when I bring them inside but cherry tom's do fine.

Most of the herbs should do fine except the rosemary - in my experience it hates living indoors.

Garlic is a seasonal plant that you plant at one point in the year and then harvest half a year later. It is not something that you can harvest off of all year long (there will be a plant but there won't be a bulb). Garlic chives should do fine in a pot indoors though and the flavor is similar.

Asparagus will be tricky since the plants get 6 feet tall and about 4 feet wide and they pretty much like full blazing sun - I move them to the outside plant list.

Carrots might work but they take more than one season to size up for me. You could grow seedlings all winter that you could then plant outdoors in the spring.

Orange trees and other citrus would work and most will stay small when grown outside of the tropics. They can get buggy but if you move them back outdoors in the summer you may not have a problem. Most citrus bloom during the winter or very early spring and then it take all year for the fruit to form and ripen - making them a great winter time crop when no other fruit ready to harvest. They don't need high heat (I keep mine just above freezing!) but they do like higher humidity than what most people have in their homes.