If a notebook does not boot up on mains power does it mean the fault lies elsewhere than the battery pack? Could it be the coin battery?
Could be either the battery pack, or the power supply itself... If the "coin battery", as you call it, were dead the unit would fire up and go into CMOS for setup because all the settings would be dumped including date and time... It wouldn't boot into the OS normally, but would get into setup.
It appears ti be a worn power socket. I cannot find it as a separate part listed with HP so could is be part of the power on button?
http://www.ebay.com.au/itm/NEW-DC-POWER-JACK-SOCKET-HP-Mini-2100-2133-2140-5101-5102-5103-/251314904869?hash=item3a83891f25:g:gW0AAOxy14VRVLrU