How I fixed my Nike+ app.
You can't uninstall, you can't reset your profile and as long as your pin is valid on the Nike server it will continue to log your workouts to some diconnected account on their end. To fix this, you have to get your iPhone to invalidate your account.
Install the trial version of Charles http://www.charlesproxy.com/download/
Download the Charles certs and install them on your iPhone http://www.charlesproxy.com/ssl.zip
Create a local map in Charles from
https://nikeplus.nike.com/nikeplus/v1/services/device/get_pin_status.jsp to a local file with the following content.
Edit your iPhone WIFI settings to use your computer as a proxy.
Sync your workout as normal. It will take a little longer because it is generating a new pin but it should now be associated with your account.