Progress! I tried adding a 2N3904 NPN transistor to switch more power into the IR LED signal, and it worked! I was able to toggle our television's mute from all the way across our family room. I did have to aim it pretty directly though, so the next thing I'll probably try is using 2 or 3 LEDs together, in order to get more spread in the beam. I might also try using a 2N2222 transistor, since it can deliver more current than the 3904.