A word for delight at someone else's failure? [duplicate]

There is an English word for this - epicaricacy. However, it is not a widely known or used word. I would say that more English speakers would be familiar with the German borrow-word schadenfreude, but even that may be limited to well-read persons.

A person who derives enjoyment from the suffering of others is a sadist, and while this carries a very strong and serious meaning, it is possible to use it in a lighthearted, softer way, for example:

He derived a certain sadistic pleasure from the misfortune of others.

Although sadism involves the inflicting of misfortune on others, the term "a certain sadistic pleasure" is used in a more vicarious way such as this quote from the NY Times:

...some of the people are so unpleasant that there's a certain sadistic pleasure in witnessing the awful things that happen to them.

However, I think the best word to fit into your sentence is "amusement":

No matter how I tried, I couldn't complete the seemingly simple task. So feeling like an idiot I gave up and asked the expert. But to my amusement, he couldn't do it either!

You do not really need the word to carry the meaning of epicaricacy because your sentence makes it quite clear what you are deriving amusement from. Your statement is more a definition of this kind of pleasure.

I'm thinking about the word gloat:

To feel or express great pleasure or satisfaction because of your own success or good luck, or someone else's failure or bad luck.

• A disaster for the media, but worth a gloat from everyone else

• I know I shouldn't gloat, but it really serves him right.

• His enemies were quick to gloat at his humiliation.

Stick with delight if you just want to fill the blank.

No matter how I tried, I couldn't complete the seemingly simple task. So feeling like an idiot I gave up and asked the expert. But to my delight he couldn't do it either!

If you want to make it explicit that you did not share this delight with anybody in order to save embarassing the expert, that's "secret delight".

As others point out this is close to the loan-word "Schadenfreude". However, it does not fit exactly in this context, and may also be missing an essential element, that the expert's failure is a misfortune for him. This instance may be more a matter of self-justification. He has proved to you that you were not the idiot that you were accusing yourself of being, out of insecurity or unfamiliarity with the work. You are, at least in this instance, every bit as capable as he is, and your self-confidence is bolstered.