I do not see why you assert that $x\wedge y$ must be either $x$ or $y$. This is not true in an arbitrary lattice: for instance, in the lattice of subsets of a given set $X$ (with $\leq$ corresponding to inclusin, $\wedge$ corresponding to intersection, and $\vee$ corresponding to union) it is false that $x\cap y$ must be either $x$ or $y$. So you should really be feeling not comfortable about your assertions well before you actually do. But even if you got to that step, you are definitely in trouble: it does not follow that $y\leq x$ "as $L$ is a partially ordered set". That assertion has no justification whatsoever.
As to your "proof by contradiction", a proof by contradiction does not begin by negating your "definition" (as you claim earlier). A proof by contradiction begins by negating the proposed conclusion. If you wanted to do this by contradiction, you would have to being by assuming that your conclusion, "$x\wedge y \leq x$ and $x\wedge y \leq y$" is false. So you would need to assume that "$x\wedge y\not\leq x$ OR $x\wedge y \not\leq y$" is true.
And even if you get to your third line, you have it exactly backwards: it does not follow that $x\leq x\wedge y$ and $y\leq x\wedge y$. What makes you think that?
So... how do you prove this?
You say you are defining the lattince operations in terms of the partial order. That is, you "know" what $\leq$ means, and then you define $\wedge$ and $\vee$ using $\leq$. The definition would be that $x\wedge y$ is the greatest lower bound of $\{x,y\}$ (which must exist if you call the poset a lattice) and $x\vee y$ is the least upper bound of $\{x,y\}$.
Well, using that definition, $x\wedge y$ is the greatest lower bound of the set $\{x,y\}$. In particular, it is a lower bound of $\{x,y\}$. What is the definition of "lower bound"? What does that tell you?
It is also possible that it is the other way around: you "know" what $\wedge$ and $\vee$ mean, and you define a partial order in terms of them. If you define $\leq$ in terms of the lattice operations, then, explicitly, you define the partial order associated to the lattice by saying that for $a,b\in L$,
$$a\leq b \Longleftrightarrow a\wedge b=a,$$
or else you define it by
$$a \leq b \Longleftrightarrow a\vee b = b.$$
In fact, both definitions are equivalent, as can be shown using the properties of the two operations in a lattice.
So in this case, what you want to show is that $(x\wedge y)\vee x = x$ and that $(x\wedge y)\vee y = y$. (Or if you defined it using $\wedge$, that $(x\wedge y)\wedge x = x\wedge y$ and that $(x\wedge y)\wedge y = x\wedge y$). These will follow directly from the properties that operations that make up a lattice must satisfy (commutativity, associativity, idempotency, and the two absorption laws).