Increasing distance of IR remote / transmitter
I assume that the led you purchased just isn't as bright as the led that came with your remote.
As I've seen from your link you could quite easily swap the led from the board.
If the new led has the same voltage, the board should work just fine with any other led. Maybe just search for a brighter led with the same wavelength.

I don't think that a back reflector would have such a big effect to boost the range to the distance you need. Also I think it would be rather complicated to install one on the tiny board.