We hacked a lot of this out in comments [above at present but may not last forever] so here's a précis of our conclusions.
Firstly, the laptop is expecting to see a headset with mic in & stereo speakers out.
Plugging in a mono jack is going to confuse the system, as impedances are not going to match the range it should be expecting from a headset. [The best guess as to why it worked on another machine is that the other machine may have had a switchable mic/line input - as a hard/software combination; we may never know.]
Replacing the mono jack with even a correctly wired TRS plug is still not going to be successful because of the massive difference in levels output by the 'line out' of a guitar amp.
Consumer & semi-pro levels can broadly be described as mic, instrument & line. Each has different impedance & voltage requirements, though many interfaces can handle all 3 on the same socket, they have hardware switching & level-matching controls to prepare them for each type. A consumer laptop or generic €5 "USB sound card" cannot handle a line output to its mic input; the voltages are just too mis-matched.
This really leaves 'buying proper stuff' as the only truly valid option.
To decide which 'proper stuff' to buy, a music shop is going to be the best start point. I posted a link to Thomann, simply because they're one of the EU's biggest box shifters so they have a lot of choice [no recommendation or affiliation].
Range of USB "sound cards" - input/output devices - from €10 to €6000…
Rather than buying one dirt cheap single-purpose device just for guitar at €15 & have to spend twice as much again later, it makes sense to anticipate future needs & look for something with line, instrument & mic with phantom power at around €30 - Something like this Behringer, a distinctly entry-level price point for this type of device. You might also be tempted for a little more to look at a 2-channel device, in case you anticipate stereo inputs at any point in future.
Once equipped with the necessary hardware, you will find your experience improves further by not using Windows' [or any other OS's] built-in sound drivers. If you use 'pro' software, Cubase, Logic etc then this will be handled by that software itself. For the lower-budget end of the market there is such as ASIO4ALL which will enable you to achieve lower latency [delay] in your audio chain.