Why did the King of Bohemia seek Sherlock Holmes's help in "A Scandal in Bohemia"?

Irene Adler has a picture of herself and the King of Bohemia, and he wants Holmes to get it back.

The King of Bohemia explains to Holmes that when he was only a prince he made a bad decision.  This decision was to have an improper relationship with Irene Adler.   Now that he is king, she is using it to blackmail him.

“… Your Majesty, as I understand, became entangled with this young person, wrote her some compromising letters, and is now desirous of getting those letters back."

At first, Holmes is not too concerned. He doesn’t see how Irene Adler can prove anything.  He says that the king can just say the letters are forged.  The issue, of course, is the photograph.  This is actual proof.

"My photograph."

"Bought."

"We were both in the photograph."

"Oh, dear! That is very bad! Your Majesty has indeed committed an indiscretion."

"I was mad -- insane."

"You have compromised yourself seriously."

Holmes has to get the picture back from Irene Adler.  Irene Adler is not just any woman.  In fact, Watson tells us that she is “the woman” to Holmes.  In her, he sees his intellectual match.  Holmes is very impressed with every feature of Irene Adler!

Five other attempts were made to get the picture before the king hired Holmes.  She is threatening to send the picture to the king’s future wife, and he says she will do it because she has “a soul of steel.”  His family will not approve, and it will ruin the marriage.

Watson doesn’t see the problem.  Holmes is Holmes, after all.  Is there anything he can’t do?  Watson says, “So accustomed was I to his invariable success that the very possibility of his failing had ceased to enter into my head.”  Holmes plans an ingenious attempt to get the picture involving setting a fire so she will have to get it out, and he will see it.

It turns out Irene Adler has heard of Sherlock Holmes.  She is a master of disguise and highly intelligent.  However, she is impressed with Holmes’s attempts to get the picture, and she gives him the picture.  What is Holmes’s payment for this remarkable case?  It is the picture of Irene Adler.