The handle isn't bringing the stopper down....hoe to fix it?

Please remove the lid from the tank on the back of the toilet to see wherein lies the problem.
You may find the flapper stuck in the upward position.
The flapper is usually black rubber, and is supposed to cover the drain hole in the middle of the bottom of the tank.
Flappers can turn your hands black if you touch them. Not a joke.
See if simple jiggling the handle will cause the flapper to work properly.
The regulator mechanism can also be worn out. That is the assembly that refills the tank after it is flushed. That can also stick and the water keeps running.
You may determine which of these is your problem.
God bless your efforts.

I struggled myself for a while looking at many online videos, replaced the flapper and the toilet still flushed on it's own! I noticed that the flapper chain got often stuck when the flapper lowered down, which caused water still running out and than I came up with my own idea; there is a white plastic fastener inside the water tank for the flush handle outside, that got apparently loose in time and can be even fastened manually, no tool needed! This fixed my problem! I haven't seen this tip anywhere, so I wanted to share it.

You'll have to see if the water is leaking past the flush stopper or into the overflow tube. The tube is the little pipe where water drains if the tank is filled too high. The flush stopper is a rubber flapper mounted on the bottom of the overflow tube. If water is running into the tube, then the flush valve may be stuck, so shake it up and down (gently) and hopefully it will be freed. If water is not running into the tube, then reach down and try to seat the flapper stopper to see if something was lodged there to keep it from seating all the way. Would be a simple fix either way.

Fill the breather hole (1/8" hole) inside the flapper.This is designed to let air out so the flapper will sink. Now it floats longer letting in more water. 25%percent more in my case.

If it has a flapper in the bottom of the tank, the flapper is probably stuck open due to something restricting it or may be just worn out. Take the top of the tank and look directly down at the center. Flush the stool and you will see the flapper come up. A few seconds it should go back down and seal the outlet.

Attached to the Flushvalve is a rubber stopper commonly called a flapper. If the flapper is worn this would cause the problem. Also sometimes the chain that attaches the flapper to the handle gets caught and the flapper sdoes not seat right. Check this, if it is clear and not obstructed try repalcing the flapper.

I had this issue as well with my toilet. My chain was unattached and had been sucked down the drain under the stopper valve preventing the toilet stopper valve from completely closing and causing the water to continually run. After re-attaching the chain I noticed when I flushed the toilet that the handle did not rise again until I manually pushed it up. After readjusting the chain to a tighter position where the chain was not floating at all but instead holding the stopper valve in place I again flushed to check the reaction. This fixed the issue of the water continually running and also the handle not resetting itself properly.

Here is the steps I took to resolve my issue:

Step 1: Remove the lid off of the tank.

Step 2: Check to see if the chain coming from the Toilet Handle to the Stopper Valve is loose, unattached, or broken.

Step 3: Flush the toilet and check to see if the problem has been fixed. If not readjust the chain to a different position and flush again.

Step 4: If all is resolved replace the tank lid.

Other possible things to check:
Check the Float Ball to see if it's Half-way or more submerged in the water. If it is it needs to be replaced.

I hope this helps!

maybe by shortening the chain length on the flapper valve might help

adjust the flappers-- either adjust the length of the lift line-from flapper to handle and or the flapper itself not properly mounted on tank seat

Lift the tank lid off, place it somewhere safe for a moment, and look at what happens: If you're talking about jiggling the handle to get the tank to start to fill with water, then it sounds like the chain or rope is out of adjustment. It can't be TOO TIGHT or the water will leak past your flapper (will cause your dripping noise, tank water level will leak down, causing constant refills), and it cant be TOO LOOSE or the toilet won't flush correctly. Since you didn't mention trouble flushing the toilet, I think that it may be a little too TIGHT. When you jiggle the flush lever-it is supposed to pull a chain or cable to open the flapper at the bottom of the tank. Flush the tank and watch what happens-all the way through to refilling the tank with water and shutting off the water refill valve-Does the flapper get stuck on the chain or something? It is supposed to fall back down to allow the tank to refill after flushing out your toilet. Does the chain look very loose? The dripping sound you described indicates that the flapper is not sealing 100%. The flapper is very easy to replace, you just have to get the adjustment set for when you flush the toilet how sensitive you need it. You amy have to play with it a little bit before you get the adjustment correct.

Hope this helps!