How to calculate bias resistor values to drive IR diode
When I asked &quot;Is there a better design method?&quot; that was not a
rhetorical question. I'd be grateful if someone could show a better
design method in an answer.

This should be a better (higher speed) driver because you are not forcing the BJT to operate in saturation (it takes vital nanoseconds to come out of saturation): -

If it's still not fast enough you might consider using a faster transistor because the BC817 is unity hFE at 100 MHz (aka \$F_T\$) - go for something like 1 GHz.
Another improvement in speed can be got by not fully turning the transistor off i.e. biasing the base a bit so that the IR LED receives (say) 3 or 4 mA when the logic input is at 0 volts. This is similar to how laser diodes are driven - full light extinction is prevented.
I also see nothing in the IR LED data sheet that tells you how quick it can turn on and off - this would significantly concern me. I see that it has a capacitance of 90 pF and that's OK when turning on the LED but not so great when turning it off.