To begin with Lash knew the ritual for the Darkhallow, not Harry. Sans Lash, why would you assume that Harry could do it, not withstanding the broken back, which would seem to be a major impediment? In a similar vein, the coin could be summoned because the Shadow knew how to do it, lacking the Shadow why do you think that Harry could summon it?
...
In the final analysis -- because Jim could write it that way, if he had wanted to. For Doylist reasons, this is sufficient unto itself! ;-)
But there are solid Watsonian lines of argument, too...
Grevane, IIRC, just skimmed the Word, and reported the Darkhallow as remarkably simple. Harry (not Lash) summoned Sue, so Harry may not even have needed Lash's memories to reconstruct such a "remarkably simple" process -- particularly since Harry seems to be "a natural" with Necromancy!
The canonical truth is: Lasciel, the Fallen Angel, made a great big STOMP onto Harry's mind. She filled it with her Shadow, and imbued that Shadow with a bunch of knowledge (Etruscan/Whamp & Sumerian/Ghoul languages (and likely others, at a guess), various kinds of magic, the fact (and implications) of Harry's Starborn-ness, etc etc etc), and at least a couple of mental quirks: a "photographic memory," and some musical techniques. Potentially, a bunch of other stuff Jim may have considered (but decided not to whip out of that particular
diabolus ex machina).
It is equally canonical that -- once enSTOMPed -- this was all IN
HARRY'S MIND, not any separate outside source. Lasciel seemed to have put up some barriers, so the Shadow would be Harry's "gatekeeper" to dole out goodies, to tempt with Easy Solutions, etc etc etc. But the Shadow itself got tempted, instead... Lash seems to have abandoned all that barrier-to-knowlege.
We know Lash left Harry some music.
We know she did NOT leave him with the exotic languages.
We know Lash left Harry with a Li'l Surprise (or maybe
became that Surprise?)
In principle, Lash could have left Harry with some/all of the Word of Kemmler, or just the Darkhallow. She could also have left Harry with instructions on Coin-summoning (it's ALSO apparently pretty easy (I guess Jim thinks "Cheap Cosmic Power" is typically made pretty easy by the Bad Guys).