According to P36 of the user manual, pressing the input button should turn the TV on from standby; does it? P5 says you have an input button on the right hand side of the TV, second from the bottom; does that work? Ignore what I said about selecting inputs using the menu; some models, "Input" is an option on the main menu as well; doesn't seem applicable to your model.
If the button on the side works, but the one on the remote won't even turn on the TV, it's probably just a broken remote. They're not normally worth spending much effort fixing; I see other LG remotes on amazon for under £20. Somewhere, there'll be a part number (maybe starting AK); try feeding that into google shopping and see what you come up with. Alternately, something more generic like the logitech harmony 300...