Running a timer through script?

You shouldn't connect to an event multiple times without reason. Consider this script instead:

--Setup variables
local running = false

game:GetService("UserInputService").InputBegan:Connect(function(inputObject, gameProcessedEvent)
    if inputObject.KeyCode == Enum.KeyCode.LeftShift then
        running = true
    end
end)
game:GetService("UserInputService").InputEnded:Connect(function(inputObject, gameProcessedEvent)
    if inputObject.KeyCode == Enum.KeyCode.LeftShift and running then
        running = false
    end
end)

If you wanted to use a wait, you would set it up like this:

--Setup variables
local running = false
local runLimit = 15 -- in seconds
local runInstance = 0

game:GetService("UserInputService").InputBegan:Connect(function(inputObject, gameProcessedEvent)
    if inputObject.KeyCode == Enum.KeyCode.LeftShift then
        running = true
        -- Make sure that if the player starts running again, this thread won't do anything
        runInstance = runInstance + 1
        local instance = runInstance
        wait(runLimit)
        if running and runInstance == instance then
            running = false
        end -- else the player has either stopped running OR stopped & resumed, meaning there's a new 'wait(runLimit)' running, meaning we shouldn't do anything
    end
end)
game:GetService("UserInputService").InputEnded:Connect(function(inputObject, gameProcessedEvent)
    if inputObject.KeyCode == Enum.KeyCode.LeftShift and running then
        running = false
    end
end)

Of course, this simply means that a player need only release 'shift' for a moment to continue running. If you wanted stamina, you have two choices:

1. (Easy) Decrease stamina every second unless runInstance ~= instance or not running, in which case break the loop. If stamina runs out, stop running and then increase the stamina after every second until runInstance ~= instance or running or until you hit the maximum stamina.
2. (Harder) Decrease stamina based on the time spent running.

Another note: you should probably have functions for "StartRunning" and "StopRunning"; these can change the running variable and also keep the player's Humanoid's speed up-to-date.