Anybody know anything about Horseradish?

Not sure this is going to help.

Horseradish is good in orchards because it keeps some of the bugs away. It also helps to deter Colorado Beetle from potatoes but I don't think you would have that problem.

As others have said, persistence does win out in the end but it will take a couple of years for you to eradicate it. I'd use Glyphosate on new leafy growth (a couple of applications three weeks apart) and then more digging.

Checking my old notes on herbicides.

Glyphosate (Roundup) is pretty good as a control and does not require a licence. It needs to be put on new leaf growth so you would have to wait for it to regrow with nice new leaves. The bigger the leaf surface the better the uptake.

Hexazinone is quite effective but not recommended where you may have crops. I don't know whether it's available anymore - and may need a licence.

2,4-D (I think a licence is needed) is a very effective foliar herbicide and doesn't contain much 'Agent Orange' anymore.

Imazapyr is also very effective but I don't know if it's available in this country - or whether it would require a licence. Not recommended to be used on areas with crops but was generally used selectively near them.

Triclopyr works similarly to Glyphosate.

My recommendation: Have chat with farmers near you and they may have good info on what is effective and available. Even if it's not available for the layman they may be able to give you some.