C172 FLT INSTRUCTOR AND STUDENT HAVE A TRACK AND HDG DEVS DURING DEP FROM SNA.

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MY INSTRUCTOR AND I GOT AN IFR CLRNC FROM SNA TO LGB TO PRACTICE MULTIPLE APCHS IN THE AREA (LGB ILS RWY 30; FUL VOR-A AND SNA ILS RWY 19R). THE CLRNC GIVEN BY SNA CLRNC ON FREQ 118.0 WAS 'CLR TO LGB; AFTER DEP TURN R HDG 220 DEGS; VECTORS TO SLI; CLB 2000 FT FREQ 124.65 AND SQUAWK. THE ATIS INFO FOR SNA WAS SHOWING 5 MI VISIBILITY; CEILING 1300 FT OVCST AND PRESSURE OF 29.95. EVERYTHING SEEMED OK AT THE RUN-UP AREA; BUT RIGHT AFTER DEP AND WHILE CLBING AND TURNING TO HDG 220 DEGS; ATC ISSUED A NEW ALT OF 1600 FT FOR THE LGB ILS APCH. BY THIS MOMENT WE WERE INSIDE THE CLOUDS AND THE FLYING INSTS: TURN COORDINATOR; HDG INDICATOR; AND ARTIFICIAL HORIZON SHOWED UP TOTALLY DIFFERENT READING AMONG THEM THAN EXPECTED AND I NOTIFIED THE INSTRUCTOR RIGHT AWAY BEFORE I STARTED TO GET DISORIENTED. WHILE TRYING TO IDENT THE PROB WE KEPT CLBING TRYING TO GET OUT OF THE CLOUDS SINCE BY WORD OF MOUTH WE HAD THE INFO THAT THE TOPS WERE AT 2000 FT. WE NOTIFIED ATC THAT WE WERE HAVING PROBS WITH THE INSTS AND WE NEEDED TO GO VFR ON TOP OF THE CLOUDS. IN THE MEANTIME; ATC REQUESTED US TO CHANGE HDG TO 330 DEGS BUT BECAUSE THE HDG INDICATOR WAS NOT WORKING PROPERLY AND WE WERE IN A CLBING ATTITUDE THE COMPASS WAS NOT GIVING US THE RIGHT LECTURE. ATC REQUESTED US TO CHANGE TO HDG 330 DEGS IMMEDIATELY; BUT BECAUSE OF THE REASONS MENTIONED PREVIOUSLY WE WERE NOT SURE OF OUR POS; SO BASED ON PREVIOUS POS WE STARTED TO TURN TO THE R AND LEVELING THE PLANE TO GET THE RIGHT LECTURE FROM THE COMPASS. ONCE ESTABLISHED IN THE 330 DEG HDG WE STARTED CLBING AGAIN. AS WE REACHED THE TOP OF THE CLOUDS AND IN VFR CONDITIONS; WE CANCELED THE IFR CLRNC WHILE WE WERE TRYING TO FIGURE OUT WHAT JUST HAS HAPPENED. ATC TOLD US THEN TO KEEP TO VFR AND FLYING TO W HDG. IT TURNS OUT THAT THE TURN COORDINATOR STOPPED WORKING AND THE HDG INDICATOR KNOB GOT STUCK IMPEDING THE HDG INDICATOR TO WORK PROPERLY. THE TURN COORDINATOR WAS COVERED AND WE FLEW FOR SOME MINS UNTIL WE WERE SURE THE HDG INDICATOR WAS WORKING PROPERLY ALONG WITH ALL THE REST OF THE INSTS.