replacing a handle flush with a push button?

My daugther has recently moved into a new home in an old house. The toilet does not flush correctly because the handle is broken. It needs replacing. The lever is situated on the front RHS of the cistern. I have noticed that the toilet has been resited, and whilst having a conventional syphon flush, when moved the overflow has been plugged and not resited. Should the cistern overfill the bathroom will flood.

Is there any product that would allow a push button flush sited in the existing lever hole and that in turn would allow overfill water to seep into the bowl rather than through an overflow pipe through the outside wall?