Here's the situation as best as I can describe it.
We have always had to hold down the trip lever of our upstairs toilet for about 3-4 seconds or else it would stop in mid-flush. A few months ago, the toilet started running and stopped flushing regularly, and would only flush when it was not running (presumably because the water got to a high enough level so that it would stop filling).
Now, the toilet runs constantly, makes noise constantly (dripping of water into the tank, and a high pitched "sssssssss" sound that can be heard throughout the upstairs rooms), and will not flush unless the tank is filled completely, and even then, the flush is not a powerful one. The flushing water leaves the bowl slowly and without much force, and when it is just about done flushing (when the water in a normal toilet swishes around in the opening at the bottom of the bowl for a few seconds), it barely glug-glugs at the bottom, leaving behind some of the old water.
The water drips into the tank very slowly. I have tried turning the water valve on to its max and it makes no difference. Just a steady, noisy drip drip. I have tried turning the water valve completely off, and the noise (and water) stops. When the tank is filled with water, the "ssssss" noise stops and all is well, and it will flush (though the flush is weak as previously described).
The problem is, the tank will not fill anymore on its own (possible need for a new flapper valve?). But if so, then why the slow dripping as opposed to a quick filling into the tank, and why the hissing?
Here is a picture of the setup: