Why wont this kill button function properly?

To kill other players, you'll need to use a remote event because of the FilteringEnabled rules.

Just a tip for the localscript, you are trying to compare a player instance with a string. On line 4, instead of using if player == v then, use if player == v.Name then, so it compares a string to a string.

For line 5, you'll need to fire off a remote event.

Then you'll need to use a serverscript (which you should put in the ServerScriptService), which retrieves the call from the client and kills the player you want.

Here's some example code.

Local Script:

local Event = game.ReplicatedStorage:WaitForChild("AdminKill") -- Waits for the remote event.

script.Parent.MouseButton1Click:Connect(function() --Kill button
	local player = script.Parent.Parent.Player.Text --Player name in the box
	if game.Players:FindFirstChild(player) then
		Event:FireServer(player)
	end
end)

*side notes: I've updated the code, the other answer was correct about it being unnecessary to cycle through each player.

Server Script

local Event = Instance.new("RemoteEvent")
Event.Parent = game.ReplicatedStorage
Event.Name = "AdminKill"

Event.OnServerEvent:Connect(function(player, target)
	if game.Players:FindFirstChild(target) then
		local targetplayer = game.Players:FindFirstChild(target)
		targetplayer.Character:FindFirstChild("Humanoid").Health = 0 -- kills the target
	end
end)

If you want it so that only certain players can kill the target players, you could make an admin-whitelist table like so:

local Event = Instance.new("RemoteEvent", game.ReplicatedStorage)
Event.Name = "AdminKill"

local Admins = {"Player1", "Player2", "Player3"}

function CheckIfAdmin(player)
	local result = false
	
	for i, v in pairs(Admins) do
		if player.Name == v then
			result = true
		end
	end
	return result
end
			
Event.OnServerEvent:Connect(function(player, target)
	if game.Players:FindFirstChild(target) then
		local isAdmin = CheckIfAdmin(player)

		if isAdmin then
			local targetplayer = game.Players:FindFirstChild(target)
			targetplayer.Character:FindFirstChild("Humanoid").Health = 0 -- kills the target
		end
	end
end)

Hope this helps! If there's an issue or you have a question, let me know in the comments. I've wrote the code via ScriptingHelpers, and not in studio, so let me know if there's any errors by testing it out)).

If you're satisfied, be sure to accept this answer)).