That is probably what I will do, utilizing the benefits of cover crop. I've heard it said that rye is really tough on weeds...hopefully the nutsedges, too. I do not have a tractor but a neighbor does...maybe he would not mind a few swipes as he does that nasty field across the street. I do have a tiller, a walk behind, might be a challenge, though. I wish I could hook up tractor equipment to the 4x4 truck.I would try mowing it as low as you can set the mower. Repeating as soon as the grass is high enough for the blade to catch much of it. Repeating multiple times. It would be most effective and most efficient to wait until the summer.
Once most of it gives up, or a few weeks before you expect winter if you get winter where you are) disk it, plant a cover crop of rye (cereal rye, not rye grass) or buckwheat. Disk that under whenever it is the best time of year to plant your flowers.
Or devote another year to a succession of cover crops. Here, I would do rye over winter, then oats, then (maybe) sudan grass or buckwheat, then rye again. Disking each under when it is about 6 inches high.
A garden cultivator would work instead of disking.
If there are any really stubborn spots, like a dip or bump that keeps the mower blade high, cover that with cardboard.

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