Let it drain, and dry out. After it dries out, maybe cut a straighter swale later, then finish with something like a box blade pulling excess dirt into the existing swale, shedding water away from what I assume is the house. Make a gentle slope working it longways with the swale, so it's easy to mow/maintain.
Some Cereal Rye mixed in with whatever grass seed you intend to plant will sprout in about 3 days after rain/watering. Mulch with straw, or pre-seeded matting to hold in place, until it gets started. Cereal Rye is deep rooting, and has quite a root mass. Mow it high for a while, 6" or so, and let it grow forming that root mass. It will penetrate that compacted soil. It will green back up after mowing, but will eventually die out later in the summer, leaving the roots to aerate the soil as they decompose. As it dies out, your other grass will fill in the area's where the Rye was.