I have been trying to keep on top of this b***ger by digging out as
much of the root as possible. But the deep tap root is so brittle its
nigh on impossible to get it out without it breaking. And it just
keeps on spreading.

Up until now I have been gardening organically (4 years in this garden)
- but I'm considering using weed killer.

Any advice on how to deal with this one?

Carpet Mulch? Weedkiller (what brand I have no idea?) Other
suggestions?