Let me sketch the argument. Philip Welch is also on
MO, and I would encourage him to post further explanation and details.
The main question left open in the original ITTM paper
was whether every clockable ordinal was writable. This was
answered beautifully by Philip Welch in
This result led to his $\lambda,\zeta,\Sigma$ theorem, which
asserts that $$L_\lambda\prec_{\Sigma_1}L_\zeta\prec_{\Sigma_2}
L_\Sigma,$$ where $\lambda$ is the supremum of the writable
ordinals, $\zeta$ is the supremum of the eventually writable
ordinals and $\Sigma$ is the supremum of the accidentally
writable ordinals. What Philip showed was that every infinite
time Turing machine computation (on writable input) repeats its
stage $\zeta$ configuration at time $\Sigma$, and from this it
follows that there are no clockable ordinals above $\lambda$.
This argument is fantastic, because it shows that whenever an
infinite time Turing machine $p$ halts on some input $x$, there is
another infinite time Turing machine that on input $x$ produces a
complete description of the computation history of $p$ on $x$, and in particular it writes an ordinal coding the ordinal halting time of that computation.
A finer analysis of the proof shows a bit more, which is relevant
for your question:
Lemma. (Welch) If an ordinal $\beta$ is clockable, then
$\beta$ is writable in time before the next gap in the clockable
ordinals.
Indeed, I recall that the stronger forms of the lemma show that
$\beta$ is clockable in time $\beta+\omega$, or even right in time
$\beta$, but for this case I'd have to think it through again, or
perhaps Philip can post about it (I don't recall which of Philip's
papers has this argument). It is interesting to note what happens
down low: in time $\omega$, we can write a real coding any
particular ordinal below $\omega_1^{CK}$, which begins the first
gap, and at time $\omega_1^{CK}+\omega$, which is clockable again,
we can write reals coding $\omega_1^{CK}$ and indeed all the
ordinals up to the second admissible ordinal.
Given this lemma, we can now answer your question.
Theorem. (Welch) If $\xi$ begins a gap in the clockable
ordinals, then $\xi$ is admissible.
Proof. Suppose that $\xi$ begins a gap in the clockable ordinals.
In particular, $\xi$ is a limit of clockable ordinals.
Furthermore, each of these ordinals is writable in time before
$\xi$, and so in particular, every ordinal below $\xi$ is writable
in time before $\xi$. Now, suppose $\xi$ were not admissible. Then
$L_\xi$ would have a $\Sigma_1$ definable map $f:\alpha\to\xi$
that was unbounded in $\xi$, for some $\alpha<\xi$, and in fact we
may assume $\alpha=\omega$ here. Thus, $L_\xi$ is the first stage
of the constructibility hierarchy where infinitely many instances
of a certain $\Sigma_1$ fact becomes true. Consider now the
infinitary algorithm that generates all writable reals, checking
if they code an ordinal $\beta$, and if they do generating a real
coding the structure $\langle L_\beta,\in\rangle$ (as in the style
of the literature mentioned above), and checking in this structure
to see how many witnesses there are to the $\Sigma_1$ fact. Since
$\xi$ begins a gap, it has sufficient closure properties that for
any particular $\beta<\xi$, all this can be checked in time less
than $\xi$. Every time we find a larger ordinal $\beta$ than
previously with a new instance of the $\Sigma_1$ fact, then we can
flash a certain master flag. By our assumption on $\xi$, it will
be first at stage $\xi$ that this master flag is on, and so our
algorithm can halt at stage $\xi$. This contradicts the assumption
that $\xi$ was not clockable. QED
