Of course, after writing my previous post I got the No Sim message and multiple resets did not fix the problem. So I called my brother who works at Apple as a genius administrator and these are the troubleshooting steps he said they take when people come into the store with this issue (only advance to the next step if the previous steps don't work):
1. Toggle airplane mode on and off (mine was fixed after this step).
2. Remove and re-insert sim card.
3. Toggle airplane mode on and off.
He said he's never seen these steps not work. He also personally speculates that something about the last update is what's causing this error (this is a personal hunch that has not been verified by Apple). If the above steps don't work, then the sim card is probably corrupt.