You could buy large rolls of black plastic, spread it over the whole mess, even up till this time next year. The soil should be solarized, killing everything. You can also use cardboard, old carpet, etc. You want to cover all of it so that when you uncover it, weeds, grass, whatever will have broken down to mulch. Then rake it through to level and start again.

Johnson grass pushes through carpet

This is a very old post, from 2006, but asks a very common question. Johnson grass is one of the invasive grass species and it spreads as much by the rhizomes as from seed. Covering Johnson grass with light excluding barriers will not stop it just as none of the poisons sold will.
The only really effective method of removal is to dig out the clumps and be sure you get all of the rhizome since more will sprout from any wee bit of rhizome left behind, just like Quack grass will.