tolkiens legendarium - From what distance can Sting detect Orcs or Goblins?

Having re-watched the Hobbit and LoTR trilogies recently and I wondered if there was any evidence in any of Tolkien's writings as to the what is the greatest range at which Sting (and other elven blades) can detect Orcs or Goblins?

In the Hobbit when Bilbo is watching Gollum kill the Goblin, Bilbo is some distance away judging from the size of Gollum on the rock, and Sting stops glowing when the goblin dies, this seems to be about 40m. Is there any known range that Tolkien provided for the enchantment?

This question asks about the frequency of updates which is similar but there isn't an answer regarding range.

Answer

Apparently about a kilometre away

In The Two Towers, Sting "gleamed dimly at the edges" while the Fellowship camped at Parth Galen. Aragorn suspected Orcs to be either on Amon Lhaw or Amon Hen.

Nonetheless as the night wore on Aragorn grew uneasy, tossing often in his sleep and waking. In the small hours he got up and came to Frodo, whose turn it was to watch.

'Why are you waking?' asked Frodo. 'It is not your watch.'

'I do not know,' answered Aragorn; 'but a shadow and a threat has been growing in my sleep. It would be well to draw your sword.'

'Why?' said Frodo. 'Are enemies at hand?'

'Let us see what Sting may show,' answered Aragorn.

Frodo then drew the elf-blade from its sheath. To his dismay the edges gleamed dimly in the night. 'Orcs!' he said. 'Not very near, and yet too near, it seems.'

'I feared as much,' said Aragorn. 'But maybe they are not on this side of the River. The light of Sting is faint, and it may point to no more than spies of Mordor roaming on the slopes of Amon Lhaw. I have never heard before of Orcs upon Amon Hen. Yet who knows what may happen in these evil days, now that Minas Tirith no longer holds secure the passages of Anduin. We must go warily tomorrow.'

_{The Lord of the Rings: The Fellowship of the Ring, Book II, Chapter X, The Breaking of the Fellowship}

From The Atlas of Middle-earth, which is a reputable source for maps on Tolkien's Legendarium.

I did a bit of measuring, and it seems that Amon Lhaw is around 2500-3500 feet (762m-1066m) away from Parth Galen, while Amon Hen is around 300-400 feet away. Aragorn felt that it was more likely for Orcs to be on Amon Lhaw, rather than on Amon Hen.

Depending on whether you want to take Aragorn's word for it, Sting seems to detect Orcs at 762-1066 metres -- about a kilometre away (Amon Lhaw from Parth Galen, not inclusive of altitude).