Remove as much snow from the roof overhanging the balcony as possible. Use a long-handled roof rake to remove snow from the top of -- and several feet behind or above -- any ice dam. This alone may adequately address the ice dam problems.
Create channels through the dam for the standing water behind the dam to exit. For immediate, though temporary relief, cut channels through the dam with warm water. Connect a hose to an indoor warm water tap or temporarily-enabled outdoor spigot and get the stream of water as close to the dam as possible. Work from the bottom of the dam to the top. This relief is only temporary, as the channels will generally fill in within a few days, so this method should be used in combination with other techniques.
Apply a suitable deicer to the dam. Avoid using corrosive materials like traditional rock salt, or sodium chloride, and instead choose a roof-safe deicer like calcium magnesium acetate. Either place small piles of deicer directly above the ice dam every few feet along the length of the dam and sprinkle lines of deicer connecting each pile to the roof edge, or fill nylon stockings with deicer and lay these across the ice dam.
Implement long-term solutions that will work to lower the roof temperatures that forced the snow from the upper portion of the roof to melt. Add insulation in the attic, insulate any stove or plumbing pipes that pass through the attic and use insulating foam spray to seal any air leaks between the attic and rooms below.