Why wasn't Arya left at the Eyrie?

It's never spelled out in the show (and the incident never happened in the books) so we are left to speculate. We also need to remember that there were two people here (Arya and Sandor) and they might have different reasons for doing what they did.

Possible Sandor motives:

• His initial desire was to ransom Arya back to her family. Robb and her mother died in the Red Wedding. Sansa is missing, whereabouts unknown. Bran and Rickon are presumed dead. Jon Snow is a penniless brother of the Night's Watch. Hoster Tully (her grandfather) is dead and uncle Edmure is a captive of the Freys. So his only recourse was to ransom her back to her last remaining relative: her Aunt Lysa. When he learns that she's dead, that's it. No more relatives to ransom her to, or even identify her. Sandor has no idea that Sansa is in the Eyrie and he has no reason to believe that Littlefinger isn't loyal to the Iron Throne and might find a use for Arya.

• He later blurts out that what he really wants is to keep Arya safe. Perhaps he's feeling guilty, or perhaps it's a result of his infatuation with Sansa. We don't know. But we do know that Sandor has no reason to think that Arya would be safe in the Eyrie. The only person he knew that might give her sanctuary, Lysa Tully, is dead. Since the Vale never joined the War of the Five Kings, he has no reason to believe that Arya has any friends or sympathizers in the vale anymore.

Possible Arya motives:

• From the start Arya never felt that going to her aunt Lysa was a good decision. They had never met before. Lysa is an unknown quantity to Arya, and she probably believes, rightly so, that Lysa would have a hard time believing the identity of who Arya really is. Arya laughing at the news of her aunt's death is probably part relief, part laughing at the absurdity of her whole situation.

• What Arya really wants is to be reunited with her last family member, her cherished half brother Jon Snow. That's why she asks the captain of the Braavosi ship to take her to the Wall. Of course, things transpired otherwise.

In addition to the possibilities iandotkelly mentioned, it's also possible that when Arya burst out laughing after Ser Donnel Waynwood informed them of Lysa Arryn's death, he refused to believe she was who she said she was and turned them away as it is not behaviour befitting a Lady.

It's worth mentioning that that scene where they arrive at the Bloody Gate is original content to the show, in A Storm of Swords, The Hound is allegedly killed by Poliver and The Tickler before they ever reach The Eyrie, so there are no explanations from there.

So the possibilities here are:

• News of Arya's arrival are communicated to Sansa. She would immediately want Arya to be ransomed, would put pressure on Petyr Baelish to do so. Given that he is trying to woo Sansa, because she is both beautiful and heir-apparent to Winterfell, he is likely to agree. Also none of this is shown on camera, which is unlikely as it would be a key plot point. Verdict: extremely implausible.

• News of Arya's arrival are communicated to Petyr Baelish. In this scenario he has several choices - he needs to be able to control Sansa, so he would choose something where he is in control of Arya's destiny. He could kill her, capture her or take her somewhere else to isolate Sansa. Alternatively he could "rescue her" from the Hound and reunite her with Sansa. However I don't believe the controlling Littlefinger would simply let her go. Again, I don't think the storytellers would leave this decision off camera. Verdict: extremely implausible.

• The Hound and Arya just assume that The Erie is not safe, turn around and leave. Even if the news that someone claiming to be Arya (and she doesn't exactly look like the daughter of a Lord) was communicated to the Erie, this would take some minutes or hours to get up to the fortress from the Black Gate. Using the Sherlock Holmes maxim of "whatever remains however improbable" - this is the only vaguely plausible course of action.

This didn't happen in the books. While the show may deviate a bit from the book, my guess is that they will not deviate in major ways. Arya meeting up with Sansa would have been a major deviation. Beside, they still needed the Hound to get injured in some fight (even though the circumstances of said fight had changed) and for Arya to leave him for dead and head off to Braavos.

Also remember that nobody in the Eyrie is supposed to know that they are actually hosting Sansa excepting Littlefinger of course and the two/three people part of the small council which interrogated Sansa following her aunt "suicide", one another was the aunt's son who was already sent away: Sansa revealed her real identity only to these people, for all the others she is supposed to be Littlefinger's niece Alayne.

Because of that the guards probably don't know about Sansa at all.

What I don't get is why the Hound doesn't enter the Eyrie (but is there some village there, anything?) just to get treatment for his human-biting wound (Jeez that scene is one of the most ridiculous of the whole show).