The J lounge has no dedicated sleeping area. (edit: IIRC there are some recliner/lounge style seats though - poolside style, not la-z-boy style- which might work for sleeping.) It is very packed at night but suggested above emptier in mornings I guess.

I don't like this lounge, dark and only a few windows into a hallway, but it has more food options than the Admirals Club. I usually connect AA to CX and do a few hours in the AC which I find more comfortable and I like the windows to the tarmac. Then I do the TBIT for just an hour or so before the flight. If you have access to the AC also, which is available airside in the AA terminal before you exit to transit to the TBIT, you might consider going there. Probably busier, but larger and a bit more comfy in my opinion.