Sorghum is quite good at suppressing weeds as it grows very tall and if planted densely will crowd out competition, is your mower a flail type mower ? That's ideal but your mower should be up to the job regardless. It is a bit late to be planting sorghum and it won't reach it's full size but may do the job. You could try Italian rye grass but the problem there is it can be hard to distinguish the rye grass from the weeds and it doesn't grow significantly higher than the weeds you are trying to get rid of. Hairy vetch is a good winter cover crop that adds nitrogen to the soil. You could try broadcasting the seed but I honestly have no idea how good germination rates would be as I use a seeder when planting cover crops, rates could probably be improved if it's possible to lightly rake or hoe the area over after. If there are too many roots in the soil to do this passing over with a kusakariki first and running the disc a centimetre or so under the soil will chop them up enough to be able to rake the area of hoe it after.
You could try an alternative plan to prepare the soil for next year, actually encouraging as many seeds to germinate as possible, then killing them with hoeing or your flame weeder and thus reducing the number of active seeds in the top layer of your soil. Then repeat the process as many times as possible. This has the benefit of actually removing those seeds from the equation so they aren't just waiting for the right conditions to spring to life next year or the year after.
You'll notice the species of weeds germinating will begin to change soon as we go into autumn. Less of the horrific summer grasses and more chickweed, vetches and other winter visitors.