Your photos are tiny. That makes it difficult to see much detail, but it looks like the rain gutter empties onto the driveway. There is no way to make a rain garden with that setup. Gutters are supposed to be slightly sloped 1/2 inch to 20 feet of run, so adding another downspout on the other end would only help during heavy downpours. The majority of rain is still going to run toward the driveway. You might need to replace the current gutter with a new one that flows toward the other corner.

Below are links to How to Build a Rain Garden and a Plant List for Texas and a Plant List for Pennsylvania that I had saved to Bookmarks. Do a search for "rain garden plant list, your city or state" for others.