I see nothing appear in my hotbar or if I open my spellbook up after delivering a killing strike.
Ok so you have 1 action 1 bonus action. Only your action can be used to take the attack action. With this it says if you kill someone, you can use your bonus action to make an attack. It might be sayin you can use your bonus action - if you still have one left - to make an attack. If youve burned the BA already, you wouldnt be able to make an attack. If you have it left, you attack, kill, and your action is gone I would imagine the attack action should show as available at that point. When youve been testing it have you been paying attention to if you have a BA or not?