My unit performed this shortened version Emma's Garden Party for the Relief Society Birthday Celebration. The abbreviated skit takes about 20 minutes to perform and uses a narrator and 14 others to read the parts of each General Relief Society President.
This skit was performed by a stake in Kearns, Utah, in 2005, but I have no other information on it's origin or history. The full version of the skit has music that relates to the era of each of the Relief Society presidents, and calls for performers to wear clothing typical of the time. This Stake borrowed clothing from Pioneer Theatre Company in Salt Lake to use for the performance.
Unfortunately, none of the sources of the stories about the Relief Society presidents are cited, and I don't have any way of verifying if they are completely accurate, but, I have no reason to doubt their authenticity.
For the full version Emma's Garden Party program, click on the relief society birthday skit link below