This is basically what happened to me. My husband also needed to travel for work. He sent the same sort of letter, and we had zero response until suddenly our passports turned up with my new visa the day before I was due to speak at the conference in question. That said, the waiting times then were typically 10 weeks, not 15 or whatever they are now.
So unless you want to go from the conference back to the US and re-apply for a visa from there, the only thing you can do is wait & hope.
(The US embassy will happily sell you an emergency second passport, but as Vicky said, using it before your application is processed will invalidate your application.)