Small jet flight instructor reported descending below charted altitude following compass failure on aircraft during approach. Training mission was cancelled and aircraft landed safely.

Narrative

The flight was the first instrument trip for a new hire Pilot Under Instruction (PUI) a former fighter pilot who had not flown analogue instruments for 20 years. The recovery was planned via the Hi-TAC to ILS for Runway XX at ZZZ under VMC conditions. Approaching hold it became apparent that neither the VOR or TACAN bearing pointers were functioning. The TACAN was restored after repeated recycling. Shortly after this HSI heading was observed to disagree with the foreflight tablet heading by 45° without a failure flag. After some fault finding the SYNCH control was used to realign the HSI and ATC informed of the equipment failure. Since it was VMC the approach was continued onto the DME arc at 3100 feet were several large birds were observed. Just prior to the lead radial the PUI started a descent which should have occurred at localizer capture. Shortly after (4') he recognized his error he climbed again to 3100' (time XA:35). At this point ATC transmitted 'Check height as an altitude alert had occurred'. Following this there was a further unflagged heading error of 45° withing 5 nm of the airfield. After this the mission was discontinued and the aircraft landed.The minimum height during the deviation was 2550 feet when the published altitude was 3100 feet. The problem occurred due to the PUI's lack of analogue experience and task saturation; additionally the compass was behaving erratically. The responsibility rested with the instructor (me) as I was over focused on looking for birds and missed the descent. On reflection I should have discontinued the training detail following the first compass failure.