The story that Hindenburg mistook Hitler for the old Kaiser seems to be legit, at least it's not doubted by the majority of Historians, most prominently Ian Kershaw (in the first volume of his Hitler biography). In my brief research on the topic, I couldn't find any hint whatsoever that Hindenburg tried to get up which I'd deem nearly impossible as his health was gravely deteriorated by that time. It was that bad that Hitler, after the brief visit, headed to Berlin immediately and, on the same night, ordered for a law to be drafted that unified the powers and duties of the Reichspräsident (Hindenburg) and the Reichskanzler (Hitler) in the case of the Reichspräsident's death. He even had all ministers (equivalent to the secretaries in the US government) to sign this law the same night. The national socialists had tried to avoid any doubt regarding their loyalty to Hindenburg in the first phase of their government, most importantly they had expressed explicitly that Hindenburg would keep all of his power with the Ermächtigungsgesetz from march 1933. They even used this "respect" to justify the new law as the title "Reichspräsident" was, in their rationale, "inseparably connected to the name of the Great Deceased" Hindenburg. Keep in mind, that was when Hindenburg was on his death bed, but still alive. The law was explicitly also signed by secretary of defense von Blomberg and made Hitler the commander-in-chief of all military, which made a coup after Hindenburg's death impossible.

On August 2nd, 9 A.M., half a day after Hitler's visit, Hindenburg died, which made Hitler officially the "Führer und Reichskanzler" with all powers of both offices.

Regarding the story that Hindenburg mistook Hitler for the Kaiser, I'm a bit torn: As I said, most Historians take this anecdote for granted, most don't even add a footnote for proof - it seems that it stems from Hitler himself, who told this anecdote later in his life and in his inner circle of trust. This makes it a bit more likely because it wasn't told for propaganda (which would've made a lot of sense to demonstrate how sick Hindenburg was and to put Hitler into a lineage of legitimate german leaders) - but it's still imaginable that Hitler just pulled this story out of his you-know-what to have a cool thing to tell.

As always, Ian Kershaws Hitler biography is a good go-to source to read up on this story and nearly all others regarding Hitler. It's a good read and written quite well.