Every Hilbert space has an orthonomal basis - using Zorn's Lemma

First I should remark that there is absolutely no need to appeal to Zorn's lemma in the case of a finite dimensional vector space.

I should also add that when talking about about a basis for a Hilbert space one has to distinguish between an algebraic space, which is a linearly independent space whose linear span is everything; and the topological basis whose span is dense in the space. The algebraic space is known as a Hamel basis whereas the topological one is known as a Schauder basis.

The use of Zorn's lemma here is quite standard, but I suppose that you have yet to see many uses of Zorn's lemma, which is why you find this to be a difficult task.

Zorn's lemma asserts that every partially ordered set in which every chain has an upper bound has a maximal element. So to use it we need to come up with a partially ordered set that has the wanted property.

In our case, this should be sets of orthonormal vectors, ordered by inclusion. If we show that the increasing union of such sets is a set of orthonormal vectors then we have shown that every chain is bounded.

The key in doing such things is to note that if $\{E_i\mid i\in I\}$ is a chain (namely for $i<j$, $E_i\subseteq E_j$) and $E=\bigcup_{i\in I}E_i$, then if $E$ had two vectors which are not orthogonal then there would be some $i\in I$ such that both these vectors are in $E_i$; since our assumption was that the elements of $E_i$ are pairwise orthogonal, this cannot happen. Similarly for normality.

Now use Zorn's lemma and assume that $E$ is a maximal element in this order. If it isn't a basis, find out a way for $E$ to be extended, which will contradict its maximality. Therefore $E$ has to be a basis, and all its elements are pairwise orthogonal, and have norm $1$.

I don't believe your argument leads anywhere fruitful. How is $H$ partially ordered that is consistent with the total order on $E$ (or $<E>$)?

Let me nudge you in the right direction: The partially ordered set $X$ you wanna apply Zorn to is the set of all orthonormal subsets of $H$, with $\subseteq$ inducing the partial order. Do you see how any totally-ordered subset of $X$ has an upper bound and how you might proceed?