So, there's the circular bulb itself, and also this little removable plastic cylinder with two metal feet on it. How do I tell which of them needs replacing, and what specification to look for to know what to replace it with?
The cylinder is called a starter, if you google for "halogen lamp starter" you'll find those. Essentially they charge up and then release a spark that ignites the gas inside the actual bulb. They are very inexpensive, so I'd start from there. Doesn't hurt to have a spare anyway.
If you unscrew it, the correct wattage should be written on the bottom between the prongs, just make sure you get one with the same rating so it works with your bulb. Feel free to post a picture of it and I can help search for one.
The prongs itself should be standardized, but the length of the starter may vary, so check if you can still close the lamp.
Essentially they charge up and then release a spark that ignites the gas inside the actual bulb.
IIRC they pull power through the inductor and the electrodes in the tube, so the inductor has plenty of current running through it and the electrodes are hot, then suddenly break that circuit. The inductor makes the voltage across the tube spike super high and an arc forms inside the tube.
The little cylinder is the switch and capacitor at the bottom of the image. It starts off closed, so power goes through the tube's electrodes but not across the tube, then opens forcing power to go through the tube.
It's the rectangular bit that actually does the charging up.
Ah thanks! Frustratingly, Bunnings closes early on Saturdays, so I'll have to wait until tomorrow to actually go out and buy anything.
I think the last time it failed I bought a pack of two starters, and I managed to track down what I think is the second one that came with that (otherwise, it's the old old starter that I never threw away...) and put that in. It doesn't seem to have helped. I also noticed, which doesn't come across well in the posted photo, parts of the bulb are noticeably dark. Is that a good sign it might be burnt out?
Yeah the dark patches on the tube are a sign that it's failed. If you get a replacement tube it would be worth trying it with the original starter (swap them back) to see if it still works.