Hi Damien,
It could either be the (visible) button that needs to be replaced or the power button switch (small circuit board). Both can be ordered, e.g. on ebay ([http://www.ebay.com/itm/Power-On-off-Volume-Button-Key-Replacement-For-LG-E960-Nexus-4-Mako-New-/111047669325?pt=US_Cell_Phone_Replacement_Parts_Tools&hash=item19daf4f64d|button] and [http://www.ebay.com/itm/Original-Power-Button-Switch-Replacement-Free-Fix-Tools-For-LG-Nexus-4-E960-PCB-/400479992142?pt=US_Cell_Phone_Replacement_Parts_Tools&hash=item5d3e77b94e|switch]).
However, before buying anything, I would check which part is actually needed.
To check the switch: Remove the switch/circuit board of the power button and lay it on the table (the contact or "eye" on top). Take a pencil and use the side of the eraser, slightly pushing on the contact several times. You need to hear a "click" or at least "feel" the "click" to be sure that the switch is still ok. If it is still ok, I would inspect the power button. Your issue seems to point to a worn out piece, either the switch or the button.
Use [guide|13050|this guide] for replacing the power button, and [guide|13041|this guide] to get access to the power switch (see last step).
Steffen