Detecting absence of current in transformer secondary
I would like to determine the moment at which all of the energy in the core has transferred into the capacitor, so I can restart the 555 timer.

There are a number of ways to check when the secondary current falls to 0. One method is to have two secondaries on your flyback transformer. A second is to insert a opto-isolator/opto-coupler into the secondary circuit, and use the output of the opto-isolator for feedback in the primary side circuit.
Without knowing how you would judge one method as better than another, I can't really tell you which is best. If your flyback transformer is already a fixed, choice, then using a second secondary is, of course ruled out.
This application note gives an example of using an optocoupler in an application very similar to yours.