Why won't it start?
My TP42 won't start either. The fan starts and two LEDs, Power and Battery are on. ThinkLight is working but the screen won't work and neither Caps Lock. The DVD drive is working, but it is useless without the screen working.
You should be able to hook it up to an external monitor before starting up, which should show any errors for startup....... ( assuming you don't hear any beeps etc).
This would be a starting point. Booting from a Linux CD might be a good test for external display depending on how the BIOS and OS is currently setup....... even if no external display is handy most people know a friend etc that has one you could use for a few min.
It sounds like the inverter or the screen is bad........ most of the time on the T40 and T60 series models i have worked on over the years the inverter had gone bad and on these models it's pretty easy and inexpensive to replace.
( If you hold a flashlight up to the screen and you can see faint text that would be a safe bet for the inverter)
As far as caps lock button not working i would wait until the other issues are fixable....... if they are, entire replacement keyboards can be found at a low price on places like eBay..... keyboard replacement is also easy to do as long as care is taken attaching the ribbon cable.
What ever parts you end up ordering, most places, esp. ebay are usually helpful in ensuring you are ordering the correct part before buying it, opting for a US based seller accepting returns is a good idea for this kind of stuff.