It sounds to me like you have identified part of the problem, which is that your soil is compacted and not allowing water to infiltrate into the ground. Lawns and turf grass often become no more valuable at infiltrating water than concrete. Typical lawn grasses have very short root systems, so they don't add much aeration and pore space in the soil. One solution to that is to add plants with deeper root systems to open up your soils and provide places for the water to pass into. A rain garden is a great example of this process. Without being on your property, it is hard to say for certain, but I am wondering if your rain garden is located properly and if it has an efficient planting. Your rain garden should be placed where the water collects, or if necessary, in the pathway of the stormwater as it moves across your property before it gets to the spot where it is collecting. When it is raining, take a close look at the path the water is following on your property, is the rain water actually making it to your rain garden? You may need to expand your garden, create a better entryway for the water into your garden, make the depression a little deeper, or add another rain garden to properly capture the water and allow it to infiltrate into the ground.

It appear that there is a gutter that flows directly into the wettest section of your yard, you could redirect that gutter or add an extension to it so that it is emptying directly into your rain garden too. You might also consider adding a rain barrel to that downspout and capturing your roof water before it reaches your lawn. Here is some information on Rain Barrels: http://extension.psu.edu/plants/gardening/eco-friendly/rain-barrels

It's also helpful for your rain garden to have plants that remain year round. Native grasses, like Little Blue Stem and woody shrubs like Winterberry would be great for that. In your picture is appears there is not much there right now. You can find a complete list of beneficial rain garden plants here: http://extension.psu.edu/plants/gardening/eco-friendly/rain-gardens/plants-rain-gardens

Finally - you may need to look at your complete property to see where all of your stormwater is coming from and where it is going. We have a great online tool you can use for that at www.stormwaterguide.org It will walk you through the process of mapping your property and guide you towards possible solutions.

Best of luck with your stormwater!