I thought I'd post this in the hope of saving some noob like me some unnecessary grief; because I have just brought two 2.2" ili9341 displays that I thought I'd fried back to life.
They were both just displaying a white screen on power-up, after failing while fiddling around with connections. While reading the ili9341 datasheet, i noticed a comment that unstable power on power-up will cause the display to go into sleep mode; also that it can be brought out of sleep mode by bringing the reset pin low for at least 10 microsecs and then high again. In desperation I hacked one of the ili9341 demo progs to do just that at the start - delay(1) was plenty after a digitalwrite(8,low) where 8 was the reset line, another write high, - and voila! a functioning display again!
Now I just have to work out a project utilising 2 spare displays!