Just following up on a couple of comments, and also a question.
Do you have access to VCDS to scan the car, to see if it's throwing any codes for the mirror. e.g. 00943 or 09944?
Regarding the mirror switch settings mentioned above, there is also an apt description to the right of heated mirror icon. i.e. they only work below a certain outside temperature. Not sure about the D4, but as an example, the heated windscreen on my D2 only kicks in when the outside temperature is below 5C.
One other comment I just came across was in relation to the spade connectors on the mirror glass. Someone had the issue of them not heating, and when they checked the insides of the spade connectors, they were full of verdigris. Once thoroughly cleaned, the heating worked again. Not sure if you had already checked these previously.
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2001 A8 D2 FL 3.7 Q - LPG Conversion, Engine: AKC, Gearbox:FBB, Colour:LY5X Aqua Blue pearl effect, Interior:WJ light beige/dark beige valcona leather with burr walnut inserts
My DIY jobs -
Replaced alarm horn,
Replaced Coolant Temp sensor,
Updated RNS-D firmware,
Installed Reversing Camera and Digital TV,
Dashcam Installation,
Retrofitted Rear Blinds,
Auto-Dimming Rear View Mirror,
Chrome Boot Struts
Sold! 
2008 S8 D3 5.2 V10 - Engine: BSM, Gearbox:KLW, Colour:LY7F Suzuka Grey Pearl Effect, Interior:Black/Black with Silver/Black valcona leather with carbon atlas inserts, Bang & Olufsen, Heated windscreen, Soft-close doors, TV, DAB, AMI, Quad Zone, Auto Boot, Auto dim Mirrors, Keyless entry, PDC with rearview camera, AFS II Headlights
My DIY jobs -
Fan Jet Windscreen Washers,
Retrofitted Electric Side Blinds