OK, I am at a complete loss.
The good news: the front camera on an iPhone 5 does not have an IR filter, so I can check on the LEDs with my phone camera after all.
I have done a great deal of searching, and everything I've seen points to the schematic I posted being the correct way to wire this setup.
When I wire the LED directly to the modulated Arduino pin, the IR receiver works within a 6-8" range.
I can confirm through my phone camera that the LED is fairly well lit.
When wired up like the schematic I posted to try and get about 100mA through each LED, the IR receiver doesn't work at all, and the IR LEDs look extremely dim through the phone camera. They are indeed lit, but it's so dim that I can only tell by pointing the camera directly at them and plugging/unplugging them to see if there's a difference.
I have tried replacing the resistor coming from the Arduino output pin to the base with a 220ohm to ensure that the base is getting enough current. No improvement.
Any ideas?
P.S. I should add that I've also tried sending 5v directly to the base pin (non-modulated) and it's not any better.