If you care to try to repair that board, a few suggestions:
Touch up the solder joints. That may be the only problem. Those transformers vibrate a little during operation at fairly high frequencies (several kHz) due to the way the circuit works, and a cold solder joint can work its way loose over time. Audible noise is not an uncommon result.
Audible noise during normal operation also happens, and the frequency and harmonics will change if you press down on the transformer since you'll change the electrical characteristics (stray capacitance, mostly). This does not necessarily indicate failure (though it may hint at faulty or incomplete design).
If got a shorted winding somewhere in there or the core has come loose, then that can be fixed. If the core is loose, you're in luck, since getting the core apart is often the hardest part of re-winding them. Just get some similar magnet wire, and re-wind the transformer exactly how it was wound before (count turns and note direction as you take the old wire off) then re-epoxy the core together. If the core hasn't come loose, you can often make it loose (to work on the transformer) by heating up the whole assembly (obviously apply heat to the ferrite core itself) via a hotplate or electric skillet until the epoxy hits the glass transition temperature, at which point it will let go.
Why yes, I do design switch mode power supplies
