If you've isolated the problem with your broken toilet to be an issue with the toilet handle, which includes the flush rod inside the tank, then your best bet in terms of fixing the problem is to replace the toilet handle with a new one. Other parts of the flushing mechanism might be to blame, however. The chain inside the tank might have become unhooked or it may have corroded and broken. If you've isolated the problem to be handle-specific, replacing it should help ensure a lasting fix.
Before starting, you will need a replacement toilet handle, an adjustable wrench, and a screwdriver. All of these items may be obtained at a local hardware store. Though it is not essential, you can turn off the water feed using the valve behind the toilet. This will prevent the tank from refilling as you make the repair.
First, remove the cover of the toilet tank. Inside the tank, you will see that the toilet handle is attached to a long metal or plastic flush rod. A chain is hooked to the end of the rod, which you will need to unhook.
Using a wrench and screwdriver, remove the lock nut attaching the toilet handle to the toilet tank. The nut is on the inside of the toilet tank. Pull out the old handle. Toilet handle lock nuts are reverse-threaded, meaning that the nut is loosened by a clockwise motion. This is opposite the way most nuts are threaded.
Insert the new handle. Make sure that the rubber washer is on the inside of the toilet tank to prevent leaking. Replace the lock nut and tighten. Be careful not to over-tighten the nut, however, as this may cause the porcelain of the toilet tank to crack.
Make sure that the flush rod can move within the toilet tank without hitting any of the other toilet parts, such as the ball float or the flush valve. If the flush rod of your new handle is metal, you can bend it slightly so as to avoid hitting any obstructions.
Next, reattach the chain to the flush rod by sticking the end of the hook through one of the holes in the the rod. The new handle will work best when there is only a small amount of slack in the chain. If you find that there is too much slack, detach the hook from the end of the chain and attach it to one of the chain's other links. Reattach the hook to the flush rod.
If you have turned the water feed to your toilet off, turn it back on and allow the tank to fill. Test the new handle. The toilet should now flush. Replace the toilet tank lid and you'll be good to "go."