What you have is called a "broiler" -- it's intended for searing the top of a dish (steak, perhaps, or a cheese layer, or a topping), not for cooking food through. As you note, anything thicker than a pizza will burn on the top before the middle is cooked, and never mind the bottom. It's primarily a radiant cooker, as opposed to the main oven which functions by heating a space and letting the heat "soak into" the food.
There isn't a good way to cook food through with a top radiant heat source like a broiler -- the only thing I can think of would be to preheat something like a pizza stone (circular terra cotta tile the size of a large pizza pan) and put the food on it, but that will only work for shorter cook times; the stone won't stay hot under the food because nothing is adding heat from the bottom.
The lower heating element in ovens (as opposed to toaster ovens, the counter top version) is a replaceable part. Unless the failure is due to a bad control, it's a pretty simple matter to pull out the dead element and replace it with a new one, and they're easily ordered on the Internet (find the data plate on the range to have the exact model number, to be sure of getting the correct part).