Well he's already got the chapters he cut from A Dance with Dragons
to put into that book, so there's something. Add to this that Storm of Swords will take two seasons (making it possible the combined Feast for Crows/Dance with Dragons will also be at least
two seasons), and if it takes him say... three or four years per book of these last two, maybe they'll make it.
Honestly one of the many big gambles of this TV adaption was adapting a still running novel series whose author had been taking longer to write each volume, I guess we'll see how that plays out in the end.
That's the thing. In GOT, he is an absolute piece of shit who throws a child from a roof and later cold-bloodedly kills Ned's men. But a few chapters told from his POV and you find yourself, despite your best efforts, liking the SOB.
I'm still rooting for Jaime's comeuppance, but he does strike just the right tragic note: Likeable enough for his POVs to have empathy in them, yet nasty enough for whatever horrible fate the books have in store for him to be well deserved.
As far as the TV show goes, I always felt they basically gave us a fair bit of A Storm of Swords
Jaime in the first season - his conversation with Ned about the murder of Ned's father and elder brother IIRC mirrors his internal monologue in that novel a bit.