Bad capacitors was a stretch - you seem to have too much working. You mention HDMI & HW render, both tied to the graphics hardware. I wonder if it _is_ booting, you just have no graphics. Does the disk access seem like a norma boot or a quick blip & stop? I might be inclined to try something like the SystemRescueCd
and boot it into VESA (low-res, lowest common denominator) graphics. If you're seeing the BIOS screen, your graphics card is booting & your LCD is working. You also might try hooking up an external monitor. If the HDMI isn't good, does it have a VGA port?
With a dead battery, you might be pulling a lot of power trying to charge it - will it boot with the battery removed?
It might just be time to cut your losses and recycle it.
Thinkpads & Toshibas have a good rep for reliability in my book, Dells, HP & Sony not so much, but that's historical, anecdotal experience and so not worth much.