The mention of the sidhe enabling faster than light travel immediately brought to mind Consorts of Heaven by Jaine Fenn, the second book of the Hidden Empire series. Published in 2009, it is certainly consistent with the OP reading it "10ish years ago".
In the story, special people called "skyfools" have the ability to create "shiftspace portals", which enable objects to move instantaneously between two points. The book deals with a child called Damaru who has this ability, and so is destined to become "a consort of the skymothers". What this means in effect is that he will be sealed into a special chamber (the transit kernel) in a spaceship, and use his special mental powers to propel it.
This is a closely guarded secret, known only to the hidden sidhe who came up with this technology:
Everyone... knew that the engines that allowed interstellar travel
were stolen Sidhe technology. It was part of the legend of humanity’s
fight-back against the Sidhe Protectorate: humans taking control of
the stars for themselves. Or so people thought. But what if the
transit-kernel at the heart of every shiftship wasn’t just black-box
technology the average spacer knew better than to tamper with? What if
the power to move ships across space without obeying the usual laws of
physics came from minds like Damaru’s?
Some people are suspicious though:
He remembered the first time Nual had experienced a shiftspace
transit. She had descended into a temporary insanity far worse than
the usual hallucinatory weirdness anyone who stayed conscious for a
transit had to endure. When he’d tried to help her, one of the things
she’d ranted about was darkness in the heart. At the time he’d put the
comment down to her disturbed state of mind - he’d just saved her
life, and she was in shock. Now he saw how her reaction could have
been due to her mind touching the mind hidden aboard his ship, the one
he had known nothing about, the one imprisoned in the ship’s drive.
This fits the main points of the question, but differs in the contents of the hidden chamber. The question states that the sidhe themselves are the imprisoned slaves, but in Consorts the sidhe are the secret rulers, who breed, search for, and enslave people with the correct mental skills.
The first book in the Empire series is called Principles of Angels which somewhat matches up with the OP's memory of "robots who were called angels and who could fly as elite soldiers". In this case, the Angels are an elite race of assassins that have implants which allow them to fly, but are not actually robots.