Maybe try adding gypsum to the soil?

I actually had a similar problem when I had my house built. They did the slope of the yard right, but right in the middle of the yard was a big dip and water would collect in that and stay there for days.

I dug a small channel from the "lake" area to the side of my house where it would drain down. I fixed it over time by putting down mulch/dirt and eventually plants. I have a banana tree near the edge of where it was so in the summer when monsoons start, it still pools a little there, but the banana gets a ton of water and it loves it. The water is gone much faster now and the channel has long been filled in. The yard still has a little dip to it in that area but the mulch that went into the soil made the water drain much faster into the ground and it's usually dry a few hours after a big rain now. The plants do really well in that area too.

Looking back, I think the fastest way to fix it is just get a lot of mulch and dump it in the area.. I think the channel was more work than necessary. Spread the mulch around and dig it into the soil, then put more over it and just let it sit.

Another option would be to put down grass, but I don't like grass much so I wouldn't do that in my yard.

Thanks - I think by adding mulch as you suggest I'll be raising the level of the (large) portion of the yard where the water pooled, allowing the water to run off into the less-compacted areas of the yard, while simultaneously making the soil healthier, worms and plants happier which perhaps will over time help with de-compacting the soil(?).