I just went through fixing the exact same symptoms on my '85 300D. Same system, with components located in slightly different locations.
Here is a great page with lots of good info on testing the components of your climate control.
http://userweb.windwireless.net/~jimc/mamerepairs.html#acctest
That black box you mentioned is commonly referred to as the blower control unit/relay. It's basically a box full of switches. When told to by other components in the climate control, it switches current between the various coils in the blower resistor (on the W123 chassis the resistor is on the inside of the front passenger fender not far behind the headlight under the hood). Normally if this relay is bad the blower will be stuck on a certain speed. If the resistor has a broken coil, however, it could potentially lead to no power to the blower.
So there are two other possibilities here - the push button panel, or the temperature control unit, which is the silver box next to the black blower relay.
The page I linked will tell you how to test that power is getting to and from the correct pins on the push button unit. If you can confirm all is well there, you can move on to the temperature control. I don't think this page has much info on testing that, but if you can verify that the push button is good, and the resistor passes visual inspection, I'd lean toward that unit. It is basically the brains of the operation, combining your desired inputs from the push button, the current state of the system, the current conditions inside the cabin, etc. and then adjusting the system to bring it all in alignment. If this unit is bad, lack of power to the blower can be on symptom.
On my car, this ended up being the problem. Swapping the temp control unit with a junkyard spare ended up bringing my blower back to life.