Diagnosing AH problems is somewhat beyond the scope of PPRuNe posts.
However, check the easy things first:
Is the gyro turning at all? Yes, if the AH indicates at all.
Is the gyro coming up to proper speed? Probably, if it works correctly in some phase of flight.
However, was it ever? Bear in mind that 3 1/8" AH and DG are not compatible with venturi tubes, they must be driven by a vacuum pump. I have seen incompatible setups, where the owner said it stopped working - no, it never worked, it just more doesn't work now.
However, gyro bearings on their way out can be a little difficult to diagnose. A first indicator would be noisy, and gyro winds down to a stop too soon following engine shutdown. If the AH gyro is not making that nice whirr, even after removal of the vacuum, something's wrong. Either it never came up to full speed (air problem), or the bearings are bad (it'll be noisy). This may be linked to a too quick spool down. Expect 5 to 7 minutes to wind down to a stop.
If you think it's an air problem, is there a DG? Is it operating properly?
If the air is good (both vacuum and filter), and the AH is not working, there is little you can do to fix it within "normal" owner capability. Yes, sending it to the shop will approach the cost of a new one. You decide which is better. If you opt for new, they may ask for your core (though I don't know why they think they should get it).
Otherwise, as A and C points out, air problems for a correctly arranged system are very easy to diagnose.