When the rebuilding of the TBIT lounges is complete (mid-2007), they will all have showers.

Meanwhile, your shower choices are only at the AA lounge in T4 (airside so you need a BP for T4 or arrive on AA). If desperate for a shower, there are several gyms that will sell a day pass for USD 10 at hotels near LAX (Hilton and a couple of others). Otherwise there is no way to get a shower at LAX these days when flying CX or most other airlines.