This used to happen to a toilet at our house years ago ... as a handy teenager still at home with my mum, this is how I fixed it.
You will need:
1 wire coathanger, like the ones you get from the dry cleaner + Something to cut it with
1 pair of longnose pliers
1. Cut a length of the coathanger to the same as the old wire. Keep it handy.
2. Get your hand underneath the mechanism in the tank that draws the water into the bowl and feel with your fingers until you find the plunger (it's the only moving part under there - you will be able to push it upwards).
3. Push it with your fingers. A long plastic pin will rise up towards the part of the flush handle that sits inside the tank.
4. The pin has a hole in it. Push your wire through that.
5. Use the pliers to bend the end into a loop.
6. Bend the other end and loop it through the flush handle.
Sorry, I'm not a plumber, I don't know the proper names of any of the nits inside the tank - but that's what I always used to do and it always worked for me!
Chris Townsend, August 2007