there are 2 important points during mirror cleaning:
1) Always keep the mirror wet if you wipe the mirror. Never use anything but air if the mirror is dry
2) If you have the mirror clean but see some minor dirt or residuals from the washing liquids just leave it the way it is. Keep in mind that a mirror has to be extremly dirty unless you really see it in the image
For cleaning I would use the following method:
In the first step use water with a detergent like it is used for dish-washing. Make the mirror wet all over and wait some minutes to solve the dirt on the mirror. Then use cotton wool to wipe very carefully, without any pressure !, across the mirror. Try to wipe from the center to the edge. If the mirror is looking towards horizon, wipe from up to down while using lots of water. Use this until the dirtparticles are removed from the mirror. Now wash the mirror with lots of distillated water. The final step is to use Aceton (CH3-CO-CH3). It will wash away the residual water drops and leave only very small aceton dropplets which dry without any residual. If you don't have Aceton you can also use compressed air to blow away the water drops from the mirror.
If you have a slight blueish stripes on the mirror after the cleaing process just leave them on the mirror. Don't ever try to remove them by wiping on a dry mirror. This can cause scratches.