How to mend a 'pop-out' sink plug thingy???

[Solo2]

The upstairs bathroom sink has one of those plugs where you push down the thing between the taps and it pops out or you pull up and it puts in the plug.

DS1 has broken it, so the mechanism doesn't work and I'm wondering if there's a missing spring/ hook that's actually flushed down the plug hole - or do I somehow just need to re-hook the plug onto the metal 'lever' there about 2.5 inches from plug entrance?

No idea how to fix it but reluctant to get a plumber out for such a small job. Can anyone - who understands what on earth I'm talking about! - and has one of those types of sink plugs, tell me how to fix it and if there's likely to be a missing bit??

Sorry don't know how to fix it.
We had one on our bath and it broke after about 6 months so DP took it out and replaced it with a normal plug and overflow-thing.

IS the plug attached to the metal lever? Ours you can just lift out the plug'n'stalk bit as one to clean it.

Has DS sheared off the small 'screwhead' shaped bit at the (bottom) end that rests ON the metal lever so the stalk hasn't got a foot to sit on? Or is it the actual level that's been broken?

If it's the first scenario it's my belief you can buy a replacement silver plug top and stalk bit from B&Q. I think you can screw the length a bit so it fits well ie opens to the correct amount when you push the tap thingy.

In the latter case, sorry, don't know what you can do about it if the metal level thing down the drain has sheared!