Finding a 'nerdy neighbour' is a good idea.

Any Linux live CD (some still fit on CD if you don't have a DVD drive) will check the hardware. If you don't mind losing any data on your hard disk try the Windows reinstallation disk; it'll probably show whether the hardware has failed. But XP SP1a (!) is ancient.

I run Linux on an 12 year old laptop but I use a very 'light' distribution (Debian with xfce). You'll need a 32bit version.

It doesn't look like a battery failure. Intel computers normally start up and ask the date and time.

Small second-hand disks are cheap at CeX - a few quid...