Perhaps surprisingly, the random variable $\ell(n)$ (with $n$ drawn uniformly from $[0,N)$) concentrates too much around $\log_2\log_2 N$ (where $\log_2$ denotes the logarithm to base $2$) to have a limiting parity probability - the variance stays bounded as $N \to \infty$, as opposed to growing to infinity.  One only recovers a limiting law when the fractional part $\{\frac{1}{2} \log_2 \log_2 N \}$ of half the double logarithm of $N$ converges to a limit, and when one does so the parity probability will usually converge to a limit that deviates slightly from $1/2$.

To simplify the calculations a little let us assume that $N$ is of the form $N = 2^{2^{2k+1}}$ (I'll leave it as an exercise to the reader to handle the general case) in the asymptotic regime $k \to \infty$.  Then the binary expansion of a randomly chosen element $n$ of $[0,N)$ consists of $2^{2k+1}$ independent Bernoulli variables (each taking $0$ and $1$ with values $1/2$).  We think of this as the initial segment of an infinite sequence of Bernoulli variables.  Now we perform the standard trick of viewing this sequence as a [renewal process][1].  After each $0$, the number of $1$s one encounters before one reaches the next $0$ is $a-1$ where $a$ is distributed according to a [geometric distribution][2] of expectation $2$. One can thus interpret this sequence as $a_1-1$ zeroes followed by a one, then $a_2-1$ zeroes followed by a one, and so forth ad infinitum, where $a_1,a_2,\dots$ are iid geometric distributions of expectation $2$.  By the law of large numbers, we see with probability $1-o(1)$ that the first $t$ for which $a_1+\dots+a_t$ exceeds $2^{2k+1}$ will lie in the range $[2^{2k}-2^{4k/3},2^{2k}+2^{4k/3}]$ (say).  Also, by symmetry we see that with probability $1-o(1)$, the maximum value of the $a_i$ for $i \leq 2^{2k}+2^{4k/3}$ will already be attained for $i \leq 2^{2k}-2^{4k/3}$.  Putting these two together, we see that with probability $1-o(1)$, $\ell(n)$ will equal $\sup_{1 \leq i \leq 2^{2k}} a_i-1$.  So asymptotically we just need to understand the distribution of $\sup_{1 \leq i \leq 2^{2k}} a_i-1$.  We have the exact formula
$$ {\bf P}( \sup_{1 \leq i \leq 2^{2k}} a_i-1 < t ) = \prod_{i=1}^{2^{2k}} {\bf P}(a_i-1 < t)$$
$$ = (1-2^{-t})^{2^{2k}}$$
for any positive integer $t$, so in particular
$$ {\bf P}( \sup_{1 \leq i \leq 2^{2k}} a_i-1 - 2k < s ) = \exp( - 2^{-s} ) + o(1)$$
for any fixed $s$.  Thus in the limit $k \to \infty$, $\ell(n) - 2k$ converges in distribution to a discrete random variable $X$ with distribution function
$$ {\bf P}( X < s ) = \exp( - 2^{-s} ).$$
(Is there a name for this sort of random variable? EDIT: it is a discrete Gumbel distribution, see update below.) The quantity $\frac{|E \cap [0,N)|}{N}$ then converges to the probability that $X$ is even, which is
$$ \sum_{j \in {\bf Z}} \exp(-2^{-2j-1}) - \exp(-2^{-2j}) = 0.4998402\dots$$
which is very slightly less than $1/2$.  (If one picked a different subsequence of $N$ one would obtain a different limit; for instance if $N = 2^{2^{2k}}$ then the same analysis would ultimately give the complementary limiting probability of $0.500157\dots$.)

UPDATE: after a tip in the comments, I'll remark that a refinement of the above analysis will eventually show that the distribution of $\ell(n)$ is asymptotic to the integer part $\lfloor \mathrm{Gumbel}(\log_2 \log_2 N, \log_2 e)\rfloor$ of a [Gumbel distribution][4], in the sense that the [Levy metric][3] (for instance) between the two distributions goes to zero as $N \to \infty$ (without any further restriction on the natural number $N$).  In retrospect this sort of answer was a natural guess, given the usual role of the Gumbel distribution in [extreme value theory][5].

Some references for further reading (gathered from following links in the comments):

<cite authors="Gordon, Louis; Schilling, Mark F.; Waterman, Michael S.">_Gordon, Louis; Schilling, Mark F.; Waterman, Michael S._, [**An extreme value theory for long head runs**](http://dx.doi.org/10.1007/BF00699107), Probab. Theory Relat. Fields 72, 279-287 (1986). [ZBL0587.60031](https://zbmath.org/?q=an:0587.60031).</cite>

<cite authors="Chakraborty, Subrata; Chakravarty, Dhrubajyoti; Mazucheli, Josmar; Bertoli, Wesley">_Chakraborty, Subrata; Chakravarty, Dhrubajyoti; Mazucheli, Josmar; Bertoli, Wesley_, A discrete analog of Gumbel distribution: properties, parameter estimation and applications,  [ZBL07482747](https://zbmath.org/?q=an:07482747).</cite>

  [1]: https://en.wikipedia.org/wiki/Renewal_theory
  [2]: https://en.wikipedia.org/wiki/Geometric_distribution
  [3]: https://en.wikipedia.org/wiki/L%C3%A9vy_metric
  [4]: https://en.wikipedia.org/wiki/Gumbel_distribution
  [5]: https://en.wikipedia.org/wiki/Extreme_value_theory