The problem with directivity of the IR led's is rather like puting the horse behind the carriage...
By using a simple lamp, as invented by Edison, meaning GLOWING, and with less current than necessary to make it light up brightly, you have an excellent 360 degrees infrared sender.
Works perfectly....

A small bulb light, like in a torch lamp glows, and if not given much current even does not produce light but MAINLY Infra red light.
So if you put a small light bulb in front of a pen, it shines in every direction!
All the problems concerning ir diodes, with their small bundle of IR light vannisch!

I do not understand why anaybody is so concernet about finding the right IR led when there are tousands of ligt bulbs available, ut to hundres of watts but allso very small ones, that shine IR in any direction, at any strengt!