Showers are in the Admiral Club in T4 ... but you have to have an onward boarding pass for travel on AA to get past security and reach the lounge. Same problem with all the lounges (they are past security and you can't reach them unless flying that day on an airline in that terminal). Alternative is to get a day room at one of the LAX hotels, for example the Four Points LAX has a 24-hour rate (no set check-in/check-out time).