Our tap handles have broken and this is what is remaining:


Ideally I would like to screw something on to what is already there and not have to turn off the water etc.
I bought these unable to see what they looked like inside in the store but I realise it's a bit more complicated than I thought.


This is the inside of the handle which broke.

All of the instructions I have found start with removing the tap cover and then unscrewing a screw in the middle of the tap, but there is no screw in ours. Will these handles fit and if so which bits do I need to remove to out these on? Will I need to turn the water off?
We are planning to replace the entire basin and tap set soon so I am looking for a quick and cheap solution for the time being.