Toilet won’t flush and found this floating in the cistern
Flush mechanisms are fairly straightforward. Diagnosing the problem is the first step. Correcting the problem may require an adjustment or a new flush mechanism.    

Diagnosis  


With the lid off of the cistern push down on the flush handle. The lever inside should be connected to the flap (usually with a chain) that releases the water at the bottom of the cistern. From the picture it appears that the connecting chain may be loose or disconnecred.  
The mystery part floating in your cistern is probably part of that connection.  
Try pulling up the flap by hand and let it release. If it flushes normally the solution is in restoring the lever to flap connection. You might be able to do that by reconnecting everything or simply replace the flush mechanism which you should be able to buy at any Home Center for a few dollars.