My car is missing in idle

A few things to consider, is the check engine light on, this would confirm that the system is seeing a fault. Are you basing the mileage loss on a physical check of the mileage, or what the trip computer is telling you? The latter has many factors contributing to it's accuracy. It could be a condition where the system is not setting a fault, because the criteria for a particular fault is not being met. The condition would have to be viewed through scan data to identify it's nature.

The fact that the light came on may be a good thing. The only problem, is that your vehicle needs scanned to extract the code(s). If it were a pre-06 Deville, you would have the luxury of an on-board feature. It would be worthwhile to find out what code the vehicle stored, and we can determine it's potential relevance to your other symptoms. Alot of the larger auto parts retailers have the capability to read the codes, and it is more than likely free of charge.