There are a few products designed for this purpose ie root-out, deep-root ect but a cheap safe and easy way would be to pour some rock salt down the hollow stem/s and fasten some polythean over the stump.
The 18" stump will be handy for leavering it out later once it's dead. Good Luck Tbird+
I recall having heard that hammering in copper nails would do the trick.
Trouble is, where nowadays does one buy copper nails?.
You could try drilling a lot of holes in the stump and filling them with creosote. Or as I did ; grow some variegated ivy over it and make it a feature in the garden
Had the same problem once and used "SBK" (Shrub and Bush Killer). It's available from good garden centres, but don't expect overnight success - it will take quite a time to die off. Also, please read the instructions on the bottle/can very carefully.
Good luck
FBG40