Footski,
Whatever they were, the components have probably been converted into SEDs.
That's Smoke Emitting Diodes, and is, as you guess, bad news
The news may not be terminal
the smell you got was the board overheating (phenolic resin - seldom a welcome smell)
The little black thingy is a diode, and the long grey thingy looks a bit like a power resistor. both or either can be killed by heat, but the diode is the more likely candidate.
However there is some reason why they have let the smoke out.
I seem to remember that you are an expert wirth a multimeter
I have one of these - or at least a first cousin - and the wire that goes into the socket is believe about 24Vac (I will check tonight).
The transformer from mains power to low voltage is a seperate device, and plugs into this "stand"
Meter set on VAC, 200vrange - plug the charger together - no battery, and try the voltage at the metal charging points where the battery should connect.
Anything? nothing?
go to DC voltage, 200VDC - do the same
Owt or Nowt?
If both give nothing, the charger stand has gone open-circuit
You could try (AC volts 200V range ) the output from the transformer at the end of the lead it should be around 22 to 27 Volts AC (not dangerous)
The "stand" is a simple and not clever device which rectifies (turns AC into DC) and feeds it into the battery (the current is probably roughly set by the resistor)
If you can find which bit has gone kerblooie we can prolly help you fix this at a distance
andrew