I ordered parts to build this thing more permanently, and they arrived today. The IR receiver is a TSOP38238, which works on 3.3 V (no more dropping diodes!), and the LED is a TSAL6400, which has known specs allowing me to calculate the right resistor to get about 20 mA (roughly 120 Ω).
Now all that's left to do is make an enclosure for it. I'll be sure to leave a nice big hole for the LED to fit through, so its leads can be bent to aim at the TV's IR receiver.