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This answer is not specific to one Logitech model, and should work for most mice. The scroll wheel on an older mouse can stop working because there is simply too much accumulated crud inside that stops the wheel from functioning correctly.
You can open the mouse by taking out the 3 or 4 screws (Phillips); they are usually located one or two under the battery(s) and another two under the glide pads under the mouse in the front. You can peel the pads off and stick them back on after reassembly. If the pads are really old and thin, then feel for the hole underneath and puncture them to remove the screws. The remaining portion of the glide pads will still work afterwards.
Now carefully open the mouse and lay the cover to the side, leaving the wires attached. Look at the scroll wheel. If crud is the problem, it will be obvious.
Now clean the crud away using tweezers and or a toothpick and then to finish off, a bit of compressed air. Reassemble the mouse. There is a very good chance that you won’t have to buy a new mouse!
I have uploaded a couple of pictures to show the problem and to show the result with the “tools” used during the cleaning process.