I assume you have tried a different cable and its the same with that? I would have thought from the video that the mini USB socket in the TomTom has broken away from the PCB and is making intermittant contact, you might be able to solder it back to the board with a fine tipped iron or possibly tape it down with some PVC tape over the top of the socket to apply pressure to the joint. I wrote this a while back when someone wanted to change a battery, but it does illustrate how to take the unit appart to get to the socket:
You will need a Torx No5 bit/ screwdriver and a pair of fine tweezers would be handy but not essential.
Remove the SD card if fitted.
Lay the device screen side down and remove the two screws from the rear of the device:
Insert your thumb nail between the silver trim and black surround near the top of the device, the main unit will start to ease from the black case:
You can now easily extract the main unit, take care not to exert force on the interconnecting cables. Note the main screen ribbon cable, under the black tape is a small PCB multi way connector, remove the tape for access.
At both ends of the connector strip is a small fragile plastic lever, these need to be eased upwards to release the ribbon cable, they only need to move a few MM in order to release the cable. The other two core plug can stay connected for this job so leave it alone.
You can now see the Torx head screws securing the main board into the housing, remove each of these, once they are all out ease the board from the housing, again the speaker wire stays in place so don't go pulling it around too much. With the main board removed turn it over to reveal the battery.
Slightly closer view of the battery and the three pin multi way plug it is fitted with.
In this last picture you can see the connector at the bottom of the PCB in the center - Mike