Shut the water supply off to the toilet and remove both the flapper and the broken flush handle from the toilet. The nut on the flush handle in most cases is reversed thread so you will have to turn it backwards to remove it. When removing the flapper I recommend wearing latex or, nitrile gloves when handling it and putting it in a plastic bag. Chemicals put in the water to keep the water safe to drink break down the rubber the flapper is made of. The deteriorating flapper can often stain your hands especially if the flapper is made of black rubber. Take the old parts with you when you go shopping for the new parts so you can match them and know the new parts will work. At the store if you will find a wide selection of flush handle available for the many different makes and models of toilets. The primary differences will be whether the flush handle is mounted on the front of a tank or, on the side of the tank, some even have a 45-degree corner where the flush handle mounts. In each of these the lever inside the tank will have a different relationship to the handle so that the lever lands close to being over the flapper. Some toilets have flush handles that are quite different such as the Mansfield toilets with their flush handle going through a loop on the flush tower. It is very important to get the flush handle that fits your toilet.
Once you get home from the store you can put the new parts into the toilet. Remember that reverse thread on the flush handle. I like to see if there is any movement in the flush handle mounting that the flush handle gets tightened in the position where the lever inside the tank ends up landing as high as it will ever get, this eliminates the possibility of the flush handle moving causing a tight flapper chain which will cause the flapper to leak. The new flush handles do not always line up properly but are close. If the lever in the tank needs its positioning to be changed I usually take a lighter and move it rapidly back and forth over a small area heating the plastic up then bend the tank lever to the position where I want it to be and hold it there while it cools. This will not affect the strength of the flush handle unless you get it too hot. Once the flush handle and flapper have been installed attach the flapper chain to the flush handle with just a small amount of slack and turn the water back on to the toilet. When the tank has refilled test flush the toilet first without the lid and then with the lid in case the tank lever is hitting the tank lid. Make sure that when you press down on the handle the flapper opens fully and stays open through the flush cycle without having to hold the handle down and that the flapper seats without leaking when the flush is complete. Make adjustments as needed to chain length as needed for proper operation.