A weedkiller will probably not kill it as you have found. Using a non-selective material such as glyphosate would be a better option. Since there are other plants, you will need to apply it carefully as you have mentioned- i.e. painting it on. You did not indicate timing- I suggest you apply it in the spring to try and knock it back, again when it is in bloom, and again fall as it is going dormant. You might have some success with pulling/digging up larger plants and then tackling the smaller ones.
