How to Kill an Ash Tree/Weed

I've had Boston Ivy on the front of my house, and it seems to have been killed by an Ash plant which has rooted itself in the area where a hole was broken in the concrete (long before my time) to plant the Boston Ivy.

Every year (for at least the last 5 years). I try to kill the Ash. I've sprayed its leaves, painted cut ends with some chemical stuff I can't remember, even (when the Ivy was already clearly dead) poured petrol on its roots.

I'm now at a loss. It keeps growing back and its roots are directly under the front wall of the house. It must definitely be causing structural damage. How can I kill it once and for all?

Thanks in advance for any assistance.