One sound, two directions
A cistern holds water because two valves are shut at the same time: the fill valve lets water in until the float reaches its level, and the flapper seals the outlet that dumps it into the bowl. A running toilet means one of those two is not shut. And here is the awkward part — both of them sound exactly the same from outside the bathroom door.
If the flapper leaks, water trickles out of the cistern into the bowl, the float drops, the fill valve opens and tops it back up, and that is your hiss. If the fill valve fails to shut, water keeps arriving and runs straight down the overflow pipe into the bowl, and that is the same hiss. In the first case the water is leaving through the bottom of the tank; in the second, it never gets to stay. Same noise, opposite fault, completely different part.
Which is why nobody should ever start this job by buying a flapper. Start with five drops of food colouring in the cistern — or soy sauce, or strong tea, anything that colours water and rinses out. Do not flush. Wait fifteen minutes. Look in the bowl. If the water in the bowl has taken the colour, water is physically passing the flapper and the flapper is your fault. If the bowl is still clear, nothing is going through the flapper and the water is going down the overflow instead. That is the whole diagnostic, it costs nothing, and it cannot be argued with.
Why the second half needs the stop valve
If the bowl stayed clear, close the isolation valve on the wall — the small screw slot or lever on the supply pipe — and watch what the water in the cistern does. If the level sits above the top of the overflow pipe, you have found it: the fill valve is shutting, but too late and too high, so the surplus simply runs away down the middle. Adjusting the float height, usually a screw on the valve body or a clip on the arm, brings the level down two or three centimetres below the overflow mouth and the noise stops.
If the level is normal but the valve kept running before you shut it, the valve is the fault — almost always a diaphragm inside it with grit or scale sitting on the seat, which stops it closing fully. On most modern valves you can take the cap off, lift the diaphragm out, rinse it, and put it back, and that is a free repair. If the rubber has perished you replace the diaphragm rather than the whole valve, which is the difference between a small part and a plumber's afternoon.
The blue block in your cistern is why the flapper failed
A flapper is a rubber seal, and rubber has one enemy in a cistern: chlorine and the solvents in bleach-based cleaning products. The blue drop-in blocks and the clip-on cistern cleaners are exactly that, held in contact with the rubber for weeks at a time, and the flapper hardens, curls at the edge and stops conforming to the seat. If you have been using them for years, you have been slowly destroying the part that keeps the water in — and the leak you are chasing is the result.
So do not replace the flapper and put the block back. Before you replace anything, run a finger round the seal face: a flapper that is only dirty with scale or a film of slime will seal again after a wipe, and that is a two-minute fix. If the rubber is stiff, has lost its shine, or is visibly curled away from the seat, it is finished and only a new one will do. And when you fit the new one, take the blue block out for good.