I personally find the idea of Sting being made in gondolin beside Glamdring and Orchrist hard. I dont have any clue as to where it was made, but i like the idea of it being created in Doriath.

'Now he drew it out. It shone pale and dim before his eyes. 'So it is an elvish blade, too,' he thought; 'and goblins are not very near, and yet not far enough.'

'(...)  But somehow he was comforted. It was rather splendid to be wearing a blade made in Gondolin for the goblin-wars of which so many songs had sung; and also he had noticed that such weapons made a great impression on goblins that came upon them suddenly.'

JRRT, The Hobbit

Probably judging from Elrond's explanation of the weapons that were found, although Elrond speaks of Glamdring and Orcrist specifically, Bilbo seems to believe his blade was also from Gondolin.