The problem with single bulb divisions is time, you need at least three pseudobulbs before the plant is big enough to flower.
So all you can do is give it good growing conditions through the warmer months so that it will grow the new pseudobulb.
Sometimes (if you are lucky) you might get two new pseudobulbs.
Find some complete Cymbidium culture notes written for GA.

Pull the bulb up out of the potting media, only have the lower 3rd of the base covered with media, at the most. You'll rot it out if you bury it lower just to keep it sitting upright. If it won't sit up in the pot on it's own, you haven't packed the media around the roots tightly enough. Cyms like lots of water spring through fall, so don't be afraid to keep it a little wetter than other orchids if it's hot out.

Keep it drier until it sprouts a new bulb and begins to form new roots, then you can increase watering slowly. Increase your light slowly if it's hot out, it can take much higher light levels once established well.

Good luck with it!

You've done everything you can at this point correctly since it has little if any roots. Eventually it will form new roots at which you can repot and allow about 70% of the bulb to show above the potting media. Give it about 50% filtered shade all year. In summer it should never be allowed to dry out so water about 2-3 times a week and fertilize. It will need to get at least 3 good size bulbs before it can flower.

For more cymbidium info go to - http://www.cymbidium.org/culture-information/cymbidium

Great, thanks for the advice and help!
I'm excited to see it develop!