Aragorn was born on March 1st. About Legolas we do not know the exact date of his birth. Elves actually don't celebrate their birthdays, but the begetting day, since bringing forth children is an act of will, so you can use this celebration instead of birthday when writing about Elves.
We do not know much about how Elves or Men in Middle-earth celebrated their birthday or begetting day, but it's very possible it was in a bit more humble way than the Hobbits, with Bilbo as the most famous example.
Birthdays are connected not only with parties, but also presents. Hobbits give them, which is mentioned as a peculiarity, so it's possible that other races receive them as we are used to. Sometimes you don't even have time to celebrate or remember your birthday, but receive a present anyways: it was Aragorn's 88th birthdays when met Gandalf the White in Fangorn again.
The deadline for this challenge is September 25th, but can be extended until the end of the month. If you need an extension to finish your story or picture, just write to the email below!
For writers: send your stories to email@example.com. The contest is anonymous and the story can't be posted publicly until the results of the challenge are announced. Aragorn or Legolas don't have to be main characters, but at least one of them must be mentioned in the story.
For artists: Submit your picture into the Teitho gallery folder of Arda-Inspired at deviantArt. The picture should be new, drawn specifically for the challenge. State that you are submitting it to the contest in the artist's comments.
Happy writing and drawing!