How to tell how sensitive a photodiode is?
Depending on the speed you need, you may well be out of luck.

You state that your current diodes "with a picosecond level rise and fall time", while you link to a PD with 12 nsec rise and fall times. Furthermore, you say that your PDs are "expensive ", but the linked units are only about 6 bucks in onesies, and if you think 6 bucks is expensive for a photodiode, I suggest you need to do some research. Try Digikey or Mouser. So it's not really clear exactly what photodiodes you are talking about.

With that said, silicon photodiodes all have pretty much the same maximum sensitivity, .55 to .65 A/W. And you only get high speed by reducing the capacitance of the chip, which means low sensitivity. So you basically have 3 options.

1) Go to a bigger PD and accept the lower speed. While you have been using very fast units, you have not actually justified your need for speed, so this may be an acceptable course of action. How much speed do you really need?

2) If you really and truly need nsec rise and fall times, keep your current diode and provide a collection lens to increase the amount of light incident on the PD. Yes, this means you'll need to redesign your apparatus, and learn some optics, but that should not be an insuperable problem.

3) Go to an APD (avalanche photodiode). These will respond to lower light levels, and you can get fairly fast units. Of course, you'll also need to learn how to drive the little buggers, but think of it as a learning experience.