How do I complete the 'Loud and Clear' quest undetected?

There's no real reason to complete this undetected (except for personal feel-goodsies, perhaps). The mercenaries guarding this place are nameless, "Mercenary" NPCs, so I wouldn't worry too much about killing them.

That said, your best bet is the same way you go about undetected in general - strip naked (armor is noisy!), hit up the dark corners, and make sure to hold shift (the default) to go extra slowly.

When I was on the island a dragon came, randomly, burning all the beehives.. um.. yeah.. also, if you have the summon dragon shout, use it.

The only person you're not allowed to kill is the owner of the house, the mercenaries are free to kill, so go ahead, you won't be at a loss of any special reward or anything.

Mercenaries are always okay to kill AFAIK

I competed this mission without killing anyone at all.

As an Argonian, I can hide underwater forever, so I went through the sewers. This allowed me to avoid the NPCs outside. The last mercenary is right in the way, however, so I shot an arrow against the far wall. He went over and I went past him to break the safe and run home. It was very effective, plus it felt like the good old Thieves Guild way!