Trying to keep an area barren with nothing growing on it just won’t work. You could kill those particular weeds but nature will always want to fill the empty space. You have a few options in my view. I don’t see that spot as unusable. You could pave it and make it a handy spot for storing stuff like hoses, garden toys, etc. You could put a storage bin there. If the end fence is blocking off more of your property you could put a gate in. Or you could cultivate it and fill the area with plants of your choosing rather than weeds. I see a trellis shadow. That could be a lovely corner.

It looks like the smother was working, but may have had holes in it, or just needed more time.

What did you have in mind for this spot after the current growth is gone?

The grass may creep back under the fence if there is not a barrier.