Going left to right on your picture you have the red float, the top of the syphon chamber (white plastic with lime-scale) and, submerged, a small white plastic oval. In the centre of this second oval there should be a rod, often black plastic, with a small hole drilled across its diameter, near the top. The metal wire from the inside black plastic part of the flush handle should go through this hole. When you press the outside handle, the inside part rises and pulls the rod up. This causes a diaphragm to force water into the syphon chamber, creating a syphon action which empties almost all the water out of the cistern and into the pan.
I would guess you have one of two problems:
a. The metal wire has simply come out from the hole in the rod. If this is the case, and you can get the rod up from its lowest level in the syphon, you can just hook them back together and it should all be OK.
b. The rod has spilt / cracked between the hole for the wire and the top of the rod. If this is the case you will have to change the entire syphon mechanism, which requires the cistern to be split from the pan and will in turn require a new "close coupled fixing kit" including a new "doughnut" washer.
If b. I'd suggest you get a plumber in considering your short timescale.