Edit - Following the edit made to the OP, I must admit it is a bit unclear what you are looking for exactly. Here is what I understand. The proof must:
- Pertain or lead to algebra
- Algebra must be harder than algebra I and II
- Be college level
- Be hard
- Not be 'related to any subject'
First, I would ask your professor for an example of something she'd find acceptable, because there are contradictions here. No college level proof is hard, no proof is unrelated to any subject.
She is basically asking you to find a proof hard enough that it is unteachable at algebra level, but at the same time, you are somehow expected to find a way to make it teachable. This sounds as absurd as asking you to find a way to teach colors to blind people.
All I can think of is the usual calculus/analysis proofs:
- Rolle's theorem
- Intermediate value theorem
- Unicity of zero theorem
- Lagrange theorem
They 'pertain to algebra' because they use the concept of functions. They would be difficult to teach to algebra students, who think of $f(x)$ as a formula, and not some abstract object. They are college level.
I suggest you read section 1.1 of Stephen Abbott's Understanding Analysis. In it, Abbott talks about G.H. Hardy and two of the proofs he used as a 'defense' of mathematics (more precisely, about how mathematics is an artistic endeavour).
The first is the proof that $\sqrt{2}$ is irrational.
The second is the proof that there are infinitely many prime numbers.
Both proofs are quite short, relatively easy to understand and although they are very old, they are still "as fresh and significant as when [they] where discovered".
So you could talk about one or both of these proofs in the first half of your presentation and then explain why they are still relevant and how easily they could be included in the high school curriculum in the second half.