Push-button and Jk flip flop
I made this small circuit plan for you:

It basically splits the signal from the button and compares it with the JKs output to determin whether J or K shall be activated. Please notice that the purpose of the two inverters in the very middle is just adding a delay. You can achieve that in a more elegant way probably in reality (e.g. loading a capacitor to the threshold voltage).
Short summary:
if Button:
if J==1:
  K=1

if K==1:
  J=1

I hope that this answer was helpful!
All the best!