If you didn't know it already, London councils are venal and rapacious when it comes to parking and traffic contraventions, and refuse virtually all informal representations. The do this to make lost of money because they know that if they refuse all reps, most people, like >95% just cough-up. Even at the discount, councils get many millions from this practice.
You could wait for the Notice to Owner and submit formal reps, but at this point, the discount option has gone. Of course they may re-offer it when refusing for the second time. The problem is that the reason your wife got the PCN is down to her error, not the councils. Yes they got the parking fee money, but as I said they are venal and also ruthless.
Others may have alternative views, so wait a bit.
