If there is a point on the property where the elevation is at least 12" below the low point of the ponding area, I would trench and install a drain to bleed the water to the low point - hopefully not interfering with your neighbor. Another idea is to create a retention basin exactly where the water wants to settle and make it more aesthetically pleasing like a rain garden planted with plants that can tolerate moist and or wet conditions from time to time. Do a Google search for "rain gardens" for ideas.

There is no place within the circle that is lower than where the water has already drained. I'm surprised the asphalt contractor didn't suggest a culvert or small pipe under the driveway shown in picture #2 . As it is, you have a part time pond, known as a vernal pond. Those trees in standing water have a probable chance of dying whether you make that area in to a rain garden or not.

You may be able to find a company that can 'shoot' or drill a horizontal drain under your driveway to the lower, other side. I assume it's lower. The pipe does not have to be very big in diameter because your flow is small.

A shallow swale should be dug to connect the standing water in picture #5 to picture #1 and 2 so it will drain and give better drainage to the existing trees impacted by grade change and standing water.

With very young kids I don't think a rain garden is a good idea at this time and where you will have to put it will kill several trees.

I have several rain gardens and they work for me. They're full of water in the winter and I mow them in the summer. I used a bulldozer to make them. The ultimate Tonka toy!

Mike

The grass in under water in the Winter and Spring. It needed mowing when I took this picture.

Using first pic as reference, it looks like water would drain toward right. As Mike points out, you have a drive in the way with no culvert to allow for water to drain out of the ponding area. I would install such a culvert whether it could be done by boring or by cutting drive and patching. One way or another water needs path to flow out of area. French drains (rock-filled hole in ground) can always be overwhelmed by heavy rains so would avoid that solution.

Thanks for all the great information. I spoke with a professional in my area and I think we have a plan. Once it drys up we are going to regrade that area and put in at least two culverts under the driveway. One on each end. Hopefully that will allow the water to run off. Already lost a few trees so Ill get them cut down while we are at it.