AIR TAXI ACFT; A PA31 ON APCH TO LAX; HAS MANY VECTORS AND A CHANGE IN TFC TO FOLLOW AND REQUEST TO MAINTAIN A HIGH SPD. THE DIRECTIONAL GYRO IS STICKING AND HDGS ARE OFF AND IN TRYING TO SLOW THE ACFT; THE FINAL IS OVERSHOT.

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LEAVING VNY FOR A TWR ENRTE CTL FLT FOR LAX (BURL2) I WAS GIVEN VECTORS FOR A FINAL INTO LAX AND TOLD TO CLB FROM THE ASSIGNED 4000 FT TO 5000 FT. DURING THE CLB; I REQUESTED A S RWY. I WAS TOLD THAT THEY MAY BE TOO BUSY TO FIT ME IN; BUT TO STAND BY. DURING VECTORS; I WAS ASKED WHAT BEST AIRSPD I COULD GIVE TO A 3 MI FINAL. I SAID THAT I COULD TRY 150-160 KTS. I WAS GIVEN FURTHER VECTORS AND TOLD TO DSND. DURING FURTHER VECTORS; I WAS TOLD I'D BE FOLLOWING; I BELIEVE; AN MD11; WHICH WAS THEN CHANGED TO A BRASILIA. DURING THIS TIME; I HAD INADVERTENTLY MOVED THE CARD OF THE DIRECTIONAL GYRO; WHICH HAS THE TENDENCY TO STICK IN THIS PARTICULAR PLANE (OF COURSE I ONLY FOUND THIS OUT LATER). I WAS TRYING TO REALIGN MY HDG; WITHOUT MUCH SUCCESS DUE TO THE TURNS AND DSCNTS; WHEN I WAS GIVEN A FINAL VECTOR TO THE LOC RWY 24R; WHILE TRYING TO JUGGLE SLOWING THE PLANE DOWN; SORTING OUT MY HDG AND CATCH THE LOC; I OVERSHOT THE APCH. CTL TOLD ME I WAS L OF COURSE; I HAD ALREADY STARTED TO CORRECT; THEN CTL GAVE ME A 360 DEG VECTOR FOR RESEQUENCING; THE SECOND APCH WENT WITHOUT INCIDENT. I THINK SEVERAL FACTORS LED TO ME GETTING BEHIND THE CURVE: 1) THIS WAS MY FIRST APCH INTO LAX AND I WAS NOT PREPARED FOR THE AEROBATICS INVOLVED INTO GETTING TO FINAL. I'D STUDIED THE CHARTS BUT WAS SURPRISED BY THE AMOUNT OF VECTORS. 2) I WAS TOO ENTHUSIASTIC IN MY DESIRE TO PROVIDE THE CTLR WITH BEST FORWARD SPD; THIS PUT ME AT 160 KTS WHEN I SHOULD HAVE BEEN AT 120 KTS; WHICH MEANT THAT BOTH THE PLANE AND MYSELF WERE PERFORMING OUTSIDE OF THE ENVELOPE. 3) I FAILED TO REMEDY THE STUCK DIRECTIONAL GYRO CARD BEFORE BEING GIVEN THE FINAL VECTOR. I ALSO SHOULD HAVE TOLD THE CTLR I WAS HAVING TROUBLE WITH THE HDG. IT WAS VERY BUSY -- BY THE TIME I THOUGHT ABOUT TELLING HIM; IT WAS ALREADY TOO LATE. SINCE THIS HAS HAPPENED; I'VE SPOKEN TO MY CHIEF PLT AND SEVERAL OTHER PLTS WHO FLY INTO LAX AND NOW HAVE A BETTER UNDERSTANDING OF WHAT TO EXPECT.