Your ground got switched around between the first and second diagrams.
+v via current regulator
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|----- analog2 in (LED anode)
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\ / <- LED
v
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------- analog1 in (LED cathode)
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[ ]
[ ] <- current sensing resistor
[ ]
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------------ Ground
analog1 value will be proportional to current
analog2 will be the voltage on the led anode
(analog2 - analog1) value will be the voltage across the diode