The fuel gauge dropping to empty would indicate the main ignition circuit or the accessory circuit had gone down. This could be from a bad relay, or a connector pulled loose from the back of the fuse and relay block - or some other breakdown of the wiring. Have a close look under the hood at where the wires emerge from the cowling and run into the engine compartment. My Jetta had a leak from the EGR at that point, which melted the insulation on some of the wires, which then shorted together. **
Also, if you had a battery connection go bad, the car wont run just because the alternator is being turned by the engine, because (unlike generators) the alternator must have some voltage on its field coil (the inside part that spins) or it wont generate any output voltage at all. So look for a weak or corroded battery terminal, and check the length of the ground cable, all the way from the battery to where the ground cables (there are a couple of them) connect to the engine or transmission or the body of the car.
Its rare for a battery to utterly fail, but it can happen, from a cracked connector internally, between the 6 cells. This can also be intermittent - and come and go; which makes finding it about impossible. Though putting in another battery would eliminate that as the problem.
I always take a penknife and gently scrape the battery posts, and the internal diameter of the battery cable terminals. This ensures a clean connection.
If your Jetta has the same type of connectors as mine did, made of thin steel pieces - I consider them to be feeble. When one gave me trouble, I replaced it by soldering on one of the battery terminals that bolts on. The kind made of brass can be soldered. The kind made of lead will just melt if you try to solder them.
** I always install an underhood light when I get a car. But if the problem is with the battery connections itself - that light goes out. So...Its handy to carry a flashlight small enough to be held in your teeth, if you need to work on your car in darkness at the roadside.
And could the loud hum have been the heater fan? Or the radio now tuned to some default frequency like 87.5?