It is quite common for a toilet lever ( often called a toilet handle ) to break. Normally when the toilet lever does not cause the toilet to flush it is often because the cistern lever arm has broken, this is much more common than the actual toilet lever itself breaking.
This is the broken toilet lever arm.
Remove the cistern lid and place it somewhere where it will not fall over and break. There is no need to empty the cistern or isolate the water supply for this job.
Here you can clearly see that the lever has broken.
Undo the screw on the lever arm and remove the broken piece of metal.
Be careful as the metal will be sharp
Undo the retaining nut using a suitable spanner and remove the old lever.
Insert the new lever arm into the hole in the cistern.
Slide on the washer ( if there is one ) and the retaining nut.
Now slide the lever arm onto the lever.
Tighten the nut to hold the lever in place.
Ensure that the lever arm is in a straight line with the syphon hook and tighten the screw.
Refit the cistern lid- Job done!