The "Demogorgon" in Stranger Things is very attracted to even small amounts of blood. Eleven is frequently bleeding from the nose at several points in the show. Why does this not attract the "Demogorgon"?
First and most obvious possible reason is "he wasn't close enough to smell it". People get cut and bleed all the time, and yet the monster wasn't showing up every time someone cut himself while shaving, got a papercut or scraped a knee. Joyce's house and the "Mirkwood" forest seems to be his hunting ground (we could see him coming through from Upside Down multiple times in those areas) and he can sense fresh blood there more easily than anywhere else. Please also note, that Barbara was bleeding for quite long period - we can see her sitting near the pool dripping blood drop-by-drop and observing how it dissolves in the water - in contrast Eleven has barely few drops of thick blood coming out of her nose that she always quickly manages to stop.
Now bit of speculation ahead:
Second possibly reason comes from the fact that Demogorgon might be afraid of Eleven - he had a good chance to sense her when she first entered Upside Down (or as @Doug Warren pointed - the blank space that is somehow connected to the Upside-Down) and yet he didn't attack her, because he had much easier prey waiting around: Hyena won't attack lion if it can attack antelope instead - it could possibly win the fight but why bother? Again, like an animal he comes after her only when she again trespasses on his ground in the last episode (and even then its more interested in boys than her)
Third possible explanation comes from the name "Demogorgon" itself and originated in this Reddit thread : it is a creature described having two heads (you can see it clearly looking at the the miniature) and yet the monster has only one head. Whats even more meaningful, Demogorgon's head supposed to bicker between each others. And then Eleven at one point exclaims "I am the monster!" - so what if she is indeed the second Demogorgon's head? This could explain why
she disappeared after destroying/banishing the monster in last episode.
Edit: The second season of the series definitely disproves the last theory since
She survived the killing of the first Demogorgon and there are quite a few more of those (or similar ) creatures not related to her in any way.