You don’t get anything that detailed in The Hobbit. Whatever description we get of Thranduil’s halls is coming from Bilbo’s point of view, and he’s mostly focused on the rooms the dwarves are locked in, and the cellars that he uses to plan their escape. So there’s no mention of any private quarters (and certainly no mention of Legolas’s room, since Legolas himself is never mentioned in The Hobbit.)
What we do get of the layout from The Hobbit is this: Thranduil’s palace was carved into underground stone caves. The entrance was blocked by a large stone gate, which could only be reached by crossing a bridge over the Forest River. Bilbo thinks that the hallways are smaller, fresher, and much more pleasant than the hallways in the goblin caves from the Misty Mountains. There are store rooms and cellars and various rooms used to locked the dwarves up in (but since they’re scattered about the halls themselves, it seems that any designated “dungeon” was too small for 13 prisoners.) Thranduil had a throne room/hall that was much taller than the other rooms and was supported by carved stone pillars. And beneath the cellars was an underground stream that flowed into the Forest River through a water gate, which is how Bilbo and the dwarves eventually escape.
In The Unfinished Tales Tolkien says that Thranduil designed his underground palace in memory of Menegroth, the great cave-palace of Thingol and Melian from the First Age. However, since Thranduil’s people were not as skilled as Thingol’s, and since they didn’t have help from the dwarves like Thingol did, Thranduil’s halls are not nearly as magnificent as Menegroth was. However, it may be helpful to have some idea of what Menegroth looked like, when trying to imagine Thranduil’s halls: Tolkien tells us that the carvings inside were meant to mimic the natural world outside - the pillars were carved to look like beech trees, with branches and leaves and carvings of animals throughout the “scene.” The whole palace was lit by golden lanterns, and there were many fountains throughout the halls. It’s also said that nightingales lived inside the caves, but that was probably more to do with Melian than anything else.
SOURCES: The Hobbit, The Silmarillion (for a description of Menegroth), and The Unfinished Tales (“The History of Galadriel and Celeborn”)
((The above map is an excerpt from Karen Wynn Fonstad’s Atlas of Middle Earth. Other than the bridge, the gates, and the cellars/water gate, I don’t think any part of this layout is based on any concrete evidence in the text. But it is nice to have a visual aid.))