I've had a problem with pigeons hanging out on my balcony for about 4 or 5 months. I contacted my buildings management and they sent in pest control after several weeks of having a cage set up out there, we caught 2 birds and I guess pest control thought that was enough. They haven't been back since. My management isn't being very helpful about this and now my balcony is COVERED in droppings. I want my balcony back. How can I get rid of them? Can their droppings make me sick in any way? How should I go about cleaning it all off? Will rice really kill the birds? I want them gone.

You make it impossible for them to perch. Get creative. I might use chicken wire and tin snips to create a sort of 'barbed wire' to wrap loosely around the railing, and well as put out loosely on the floor. Plastic barrier type fencing might do the same.
To clean I would sprinkle bleach mixed with a little hot soapy water on the droppings, but not so much that it goes on the balcony below you. Wait 20 minutes then scrape off with an ice scraper, into a paper bag or bucket, again so as to not land on the balcony below.
Once they are gone, put a plastic owl on the railing.
Why kill some with a painful slow death, when more will just be back? You PREVENT them from liking your balcony.

I can't believe anyone thought that trapping pigeons would do anything!

Pigeon droppings can harbor a fungus and cause a lung problem for humans called histoplasmosis, but it usually only affects those who have weakened immune systems. Although there is controversy whether this really is possible. Otherwise, pigeons can sometimes be infested with mites. When pigeons are removed, but their nests are not, the remaining mites might find you to feed on.

Joy is right that exclusion is the most permanent method. There are products, like Nixalite, that are specifically designed for this purpose. If you use your imagination you can create your own exclusion methods. However, fake bird predators, like a plastic owl, and sound devices do not work. Also, if you can harass the pigeons regularly, they will stop coming round. Pigeons are not a protected species, and can be harassed any time they are causing problems.

Cleaning is cleaning. If you apply bleach like Joy recommends, let it soak a bit, to help kill pathogens. Maybe wear a dust mask to prevent breathing in particles. Here is a good link: Cleaning & Pigeons

Here are more internet resources:
Internet Center for Wildlife Damage Management
pigeon management
Pigeons and disease
eHow: Cleaning Pigeon Poop

There is a gel that can be applied to balcony railing that the pigeons don't like. Or buy yourself a supersoaker and spray them every time they land on your balcony.
Once again if you make them as uncomfortable as possible they will leave.