Looks like a gash job.
Lighting circuits (UK) are called radial circuits and the power cable runs from fitting to fitting or plate switch to late switch.
A pull switch is only needed in bathrooms and usually doesn't need three cables as the switch drop cable isn't needed.
Are all three visible cables in the junction box? Or is the lamp on it's own cable?
You see the screws and the slots? all you need to do is loosen the screws and twist the white part until it lines up with the screwheads.
The junction box behind in UK speak is called a BESA box and the screw centres are standard. Ideally, that white part should be flush fitted to the ceiling. If you get a replacement, ask for a surface mount type, not flush
You could do it if you are careful. Turn the power off and expose the wires. Note where they went. Turn the power back on and see if all the other lamps work. If it's a radial, some lamps will work and some won't. You will need a meter to see which wires are live. Measure between live and neutral or your hot and neutral. Look up which colour is live or hot. There should be power on one cable.
I take it you have checked the fuse box/consumer unit for failed fuses or circuit breakers?
If you have a meter, you can also test if the pull switch is working, but that's more complicated. Do your lamps screw in or bayont fix? If they screw, the centre conact should be the live. Test between the cntre contact and the incoming live. Use the bleeeper function on the meter and try switching the lamp. With cables disconnected of course.
If you are at all unsure how to do it, then don't. You could just replace the pull switch and see if it works. Could just be a loose wire.
$200 is why electricians have nice houses. $100 or less for a simple light fitting
