Vishay has published how to measure light density and intensity at a conical distance in a document from the datasheet page you posted. "General Information Measurement Techniques"
Other info
Beamwidth Standard
IR LEDs are always specified in half-angle from peak-centre (due to reflector centre error in the early days)
Visible LEDs are in full angle for any colour and angle.
Sharp Japan was the chief developer of IR 5mm LEDs and PD's and that business was sold to Vishay.
Silicon detectors are wide-viewing angles and follow Lambertian Cosine response much like PV power arrays.
- Vishay/Sharp Si Photo Diodes (PD) 5mm types are very accurate cheap sensors with consistent 5 mA/mW. More expensive detectors may be slightly higher. Yours might be less.
To convert the optical power to electrical power you need to match the impedance due to the Maximum Power Transfer (MPT) Theorem. This can be easily measured as Zmpt=Voc/Isc for open-circuit Voltage over Short Circuit current. In both of these states, the power is 0 but the ratio happens at the MPT operating point. This is critical to understand in any solar power design. The Vmpt is about 82% of Voc and then drops slowly with solar power to ~ 70%Voc for low power.
The LED should not exceed suggested values in the datasheet for DC and pulse-operation otherwise self-heating can degrade the results. You can run yours in series with a good active current limiter circuit and then controlled without worries. This can be done using an LM317 and a single resistor with a low-side switch.
