This seems pretty simple, and there are actually lots of options for you.
LAX-FRA = 5805 which you said was 1000 miles too short
LAX-SFO = 500
SFO-IAD = 2419
IAD-FRA = 4080
Total = 6999 so that should do it.
You can even substitute IAD-FRA with IAD-JFK-FRA or IAD-MUC-FRA; or LAX-SFO-IAD with LAX-SEA-IAD.
Try going north and south a bit if you're short some mileage, and for long haul segments, try taking the shortest possible segments and stay further south to generate more mileage (if you're above the equator).