The inverter outputs a fairly high AC supply. Its actually quite unusal for a backlight to fail entirely. Hate to be the bearer of bad news, but its more likely to be the power coming from the logic board. Its quite easy to short it out permanently. (This should be a DC supply, but don't measure it at the board end as this alone can short it out. Voice of experience.)

If the board isn't powering the inverter, its time for a new board. Or a new laptop.