It does look like its the ball valve in the toilet cistern.
Flush the toilet & hold the ball valve up to stop it filling, you will more than likely see it dripping.
To remedy fit a new diaphragm & nylon seat to the ball valve or buy a new valve.
Have a look under that ring that lifts up at the bottom of the tap there should be a grub screw, isolate the water supplies, then unscrew the grub screw & the tap should just lift off, revealing a plate underneath with a screw that you will be able to tighten with either a screwdriver or an...
The bolt kit you posted the pic for does not require any silicone dabbed anywhere near it the 2 conical seals will be sufficient.
I always think "cowboy" when I see silicone inside or under a recently installed cistern!
Check that it has just split & has not been chewed.
If it's split as said above was a cheap stiff connector, if it's been chewed then get some rat poison down.