This is a fairly common problem with the later type dipping mirrors on 2cv, Dyane, DS, etc.
This is how I solve it:
1): I get a bowl of fairly hot* water with a small amount of washing up liquid mixed in. *Not so hot that I can’t comfortably keep my hands in it!
2): I place the mirror assemble into the water and swish it around, drain it, swish it around, drain it, and so on until and little bits of dirt etc have drained out. I then let it soak for 20-30 minutes.
3): I remove it from the bowl and let it drain.
4): I then leave it somewhere to dry, (eg: airing cupboard or in the sunlight), in a position that allows any remaining water to drain out fully.
This almost always works for me and the mirror stays clear for ages. Sometimes however, there can be slight residue streak marks left on the glass. In such cases, I have redone the above with less soap to cure this but have recently heard the suggestion of rinsing with de-ionised water at the end of step 3.
If you have tried the above method, a variation of it, or something else, please let me know how you got on.