Breaking off a bolt instead of it coming out, sucks, no doubt. Theres two angles of attack, pre and post. Before trying to loosen a rusted fastener there are a couple things that will help if you have a shot at all. First is heat, propane is ok, mapp gas is hotter, oxy/acet is the balls. Heating a fastener breaks the rust bond, and after a minute of cool-down your odds are greatly increased at it spinning out like it is supposed to.
Second, if you can't apply heat due to plastic parts or other reason, is some penatrating oil and impact/vibration to jar it loose. If you don't have any air tools you can use a hammer, block of wood, and the socket. Any other tips are welcome as I'm sure many have there own recipes.
As far as getting a broken fastener out, If you can get at the back of the existing piece with vice grips and turn it as if you were tightening it, you have a shot. Probably not tho, so you are looking at drilling. If you don't want to destroy the hole/threads, try drilling a hole 3/4 the size of the broken piece, right in the center, and use a screw extractor. It's a backwards threaded boring bit that bites into the hole you drilled, but using left handed direction to hopfully pop your piece loose. In my experience these only work at best, 1/2 the time. If your bolt was so frozen in place, a screw extractor usually won't do it and may break off in the hole you drilled. The other choice is to drill the hole out and try to rethread it or end up using a nut-bolt combo if you can. Hope this helps, sorry so long.