Not really an answer, but a thread on entering the double [[ ]] pair (with the cursor between both pairs) using a single keystroke occurred a couple of weeks ago on the mathgroup. It didn't help me, but for others this was a solution apparently.
EDIT
to make good on my slightly off-topic first response here's a pattern replacement that seems to do the job (although I have difficulties myself to understand why it should be b and not b_; the latter doesn't work):
Defer[f[g[h[[i[[j[2], k[[1, m[[1, n[2]]]]]]]]]]]] /.
HoldPattern[Part[b, a_]] -> HoldPattern[b[LeftDoubleBracket]a[RightDoubleBracket]]
I leave the automation part to you.
EDIT 2
I discovered that if you add the above rule to ImportAutoReplacements and paste your SO code in a notebook in a Defer[] and evaluate this, you end up with a usable form with double brackets which can be used as input somewhere else.
EDIT 3
As remarked by Mr.Wizard invisibly below in the comments, the replacement rule isn't necessary. Defer does it on its own! Scientific progress goes "Boink", to cite Bill Watterson.
EDIT 4
The jury is still out on Defer. It has some peculiar side effects, and doesn't work well on all expressions. try the "Paste Tabular Data Palette" in the toolbag question for instance. Pasting this block of code in Defer and executing gives me this:
It worked much better in another code snippet from the same thread:
The second part is how it looks after turning it in to input by editing the output of the first block (basically, I inserted a couple of returns to restore the format). This turns it into Input. Please notice that all double brackets turned into the correct corresponding symbol, but notice also the changing position of ReleaseHold.
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