Can you advise how your plugging the SatNav into the car, are you plugging it directly into the car's USB socket or via an adaptor in the cigarette socket?
The most likely cause is a failed battery, for the unit to work the battery has to hold on to some charge, during startup a lot of energy is required, this is probably why you see it briefly flash when you try and turn it on (not sure why you dont see anything on the Audi, but that could be simply due to the time between last time you tried to turn it on compared to when you tried it in the Qashqai).
You can pick up a replacement battery for around £10 from online stores such as here, the video here shows you how to replace it, its not overly complicated (takes about 10 minutes).
Please let me know if you need anything clarified.
Just to clarify, I am talking about the battery in the Garmin Sat Nav not the car battery, have you tried leaving the sat nav pluged in (on charge) while the SatNav is switched off, unplugging it after an hour and seeing if it will turn on?
To rule out a cable \ connector issue, you could also try plugging the SatNav into the computer using the USB cable, after a couple of hours it should charge up from the computer, and see if you can turn it on that way, if that works, it suggests its the docking station or car cable at fault.
Please let me know how you get on.