How do I become a magician?

How do you deal with customers that want the impossible? How do you become a magician and satisfy them? For example:

$Me: (standard opening)

$Customer: I made an appointment yesterday but the previous agent made a mistake and set me up for the wrong timeframe! I cant be there at that time. I said (timeframe he wanted).

$Me: Ok, I will be very happy to check what happened there and look for a fix. (verifies). Ok now let me pullup our appointments to see if I can get you the one you wanted.

(a 2 minute hold or so while I try the methods I have learned over time to get something sooner on the system, couldnt find crap, soonest appointment is for tomorrow. Let me make it seem as if Im trying all I can)

$Me: Ok sir Im back and the soonest I was able to find was for tomorrow.

$Him: What? This is your fault. I clearly stated to the other agent what I wanted.

$Me: Exactly sir, it is our fault for having misunderstood you. I know this is frustrating. I want to really try all I can so let me check another list to see if I can find any appointments cancelled today. Ill put you on another brief hold

(again like 1 minute and a half or two minute hold — just thinking about all the ways this could go wrong once I tell him the soonest one with that time is tomorrow because there is actually no list like that lol)

$Me: Hello sir Im back, thanks for waiting. Unfortunately there are no cancelled appointments today but what I can do is keep an eye on it and if I find anything call you back to see if you want that sooner appointment. I will also apply a credit regardless for the trouble that you are going trough. (I credited like half of his bill)

$Him: This is crap. Ill take it anyways. (hangs up)

What do you do in cases like those when you cant give the customers what they want? Im not Houdini and this time it was worse since it was our fault completely. I would normally dont think about it too much but the thing is they leave negative surveys…