I just wanted to provide an update to those who may still be struggling with this issue. GoDaddy finally gave me a solution which seems to have resolved the problem:
1) Update to JetPack 2.4.2 (latest version).
2) Downgrade to PHP 5.2 if you were running PHP 5.3.
3) Recycle the application pool.
I don't have any other PHP applications running except for WordPress, and the version of WordPress I'm running is 3.6. This is an acceptable solution, and my hats off to the GoDaddy support staff for going the extra mile to help me find a suitable workaround.