First thing would be to check the battery, if it has been left uncharged for any length of time it is probably cactus, I know this from experience when I forget to regularly charge up one of the vehicles here. See if it holding charge after you disconnect the charger and then again if you put any load on it. A cheap multimeter will show you this.
Second one with the gearbox, if it was working previously and is now just flopping around then it may have broken the gearstick itself below the ball that is shown in your video. I can't think of anything else it may be really as the bottom of the gearstick sits in cups across three selector shafts and it would be hard for it to have slipped out of that to allow the movement you seem to have. So yes, take the floor out and have a look at the top of the box.
Hope that helps some,
Cheers,
TimJ.
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