How to make destructible AI spawner

Ive been trying to make this work for a while now, and no luck. I would like to have a spawner, lets just say a cube spawn AI constantly until its taken a certain amount of player damage. Ive managed to get cube or set actor to spawn AI but It will keep spawning AI even after destroyed. I was told that this is easy to do in Blueprints, but im having no luck at all. I was also told to just use c++ as this is something thats is a core mechanic of my game and will be needed a number of times in levels of game. Any help and which way I should go with this would help a lot thanks.

If I was going to attempt this I would check health which you may be already doing every time damage is applied and if health is <= 0 set life span =.01 and see if that works.

Hey Fantasmagore,

Destroying Blueprint Actor ought to stop it from spawning anymore mobs. Could you post an image of how you have your spawner Blueprint set up? Thanks!