As DIYISFUN said, burn it, but you will probably have to wait until the autumn, because as you have found it is relatively 'green' at the moment and so creating a lot of smoke.
The other way to do it so that you can get at the roots is to use a bow saw or pruners to cut through all the stems to about 6-12 inches. Then you can just dig out the roots. You will have a hard time of it and you will be left with a big hole which will need filled.
If you intend to grass over it, make sure you fill the hole in with good quality soil and compact in 2-3 layers. Then water it well and leave it a week to see if it sinks any further. Top up and then grass seed or turf.
No weedkiller is really going to touch it because, the leaves are so tough.
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