I would say that it is likely that the enchanted stream was a river running north from Dol Guldur. If you remember, Dol Guldur is built on high land in the south portion of Mirkwood. So it makes sense that a stream would form flowing from South to North cutting along the elves path through the forest.
Why the stream makes you fall asleep, I dont know exactly for sure. But it is because of Dol Guldur I would say.
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Son of Isildur.
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