I made this into a website that is a tad easier to use (long defunct, though). The code for that is also available. Original README follows below.
The Nike+ GPS app is great when starting to run, but it's not something for the long term. One problem with switching to other programs or websites is that you will probably want to take your data with you. While there is no official way to do this, it is possible to coax the website into surrendering your data in a JSON format. This script converts that data into the GPS Exchange Format (GPX), which most running websites and programs accept.
You will have to do this manually for every run. Go to the runs tab on the Nike+ website and click on a run you want to export. In the URL bar, you will see something like this:
Where the user-id and run-id are numerical. Use wget or curl to retrieve the following URL to a file (replacing the run-id part with your run-id):
Name the file so that it ends in .json, because the script will produce a file with the same name and a .gpx extension (and it doesn't check whether the extension is correct or not). Run the script:
./nike2gpx filename.json
The resulting .gpx file can be imported to RunKeeper, etc.
This is a very simple (and ugly) script that produces a fairly basic GPX file. I make no claims to the general correctness of the output, but RunKeeper accepts the files.
Nike2GPX was written by Robert Kosara. I am placing it in the public domain.