As it was probably the electronics that were broken, we took off the circuit board from the hard drive to examine it. Using a multimeter, we firstly connected the black cable to ground (GND), i.e. the common reference potential for all voltages on the circuit. With SATA, for example, there are the voltages +12V, +5V and +3.3V, which all refer to ground/GND. On the board, we could see that the top layer, i.e. the top copper layer, has been flooded with ground at the through-holes for the screws so that the hard disk housing is also at the same potential as the common reference voltage of the board due to the connection with the screw head. Therefore, we could be reasonably sure that we could connect the black cable there as the negative terminal. We then used the red cable of the multimeter to measure the voltage upstream of the burnt-out resistors directly at the plug (there was a voltage there) and then downstream - the voltage here was 0. To make sure that it was really due to the resistors, we then switched the multimeter from voltage measurement to continuity test and tested whether the resistors had continuity. They did not, so the circuit was open and no current was flowing. These resistors, we concluded, are probably there as an extra step of power surge protection: it seems they did their job! However, this damage doesn't come from nowhere, so switching the multimeter to the diode setting, we found out that a transient-voltage-suppression diode was broken, again probably from a power surge.

Looking at the handy-dandy diagram I drew below, you can (hopefully) see that we put a little solder on each connection of the broken diode to ease the seperation using a soldering iron. After taking off the diode, there were still the broken resistors, which we bridged using some solder and a little wire; as the resistors are running in parallel and this is just a dirty fix to get the drive running long enough to save the data, we just ran the wire between both.

Handy dandy diagram