It looks, from the picture supplied, that your assignment is to connect the 3 pins of the switch to the three solder points marked LP15, LP16 and LP17 in such a way that the switch has the following behaviour.
Position 1: display backlight ON, Alarm (LSP SD) ON
Position 2: display backlight ON, Alarm (LSP SD) OFF
Position 3: display backlight OFF, Alarm (LSP SD) ON
You see the "tricky" part as getting the display backlight to switch ON when the switch is at position 2 which you reduced to "lighting a LED at the middle position". However, it would be much easier to think of it like this: "How do I switch the display backlight OFF when the switch is in position 3?"
In principle, an NPN transistor as a low side switch for the display backlight with the base normally pulled high to keep the display backlight on. In position 3 the base is held low, switching the transistor (and backlight) off. You'll need the diodes already mentioned to avoid a direct connection between 1 and 3 on the switch, when connecting the alarm, and a couple of resistors.