If you are a WP or SG then the One World lounge finder will answer this question for any airport in the world that has a one world lounge:
oneworld - Search for a oneworld Lounge
In this case you have three lounges available if the third party JAL lounge says OK, the F & J QF/BA/CX lounges and also the JAL lounge on level 4, experience has shown the hours may not be accurate though:
British Airways, Cathay Pacific and Qantas Joint Lounges (First and Business Class)
Location details: Tom Bradley International Terminal, Departure Level 5
Opening hours: Daily 06:30-02:30
Notes: Access to First Class Lounges is restricted to Emerald members or those travelling in First Class
International Lounge
Location details: Tom Bradley International Terminal, Departure Level 4
Opening hours: Daily 06:00 to last departure
Operated by a third party on behalf of: Japan Airlines
* At peak periods, access to certain lounges may be restricted due to capacity constraints. Access is available on the day of departure when the next onward flight is with a oneworld member airline. Access may not apply at a limited number of third party lounges.