Thanks for all the help.
Well, the input voltage is 11.06V.
I have checked the current output from the 7805 last night, and if I can recall, with the backlight on, it was outputting just more than 0.7A!
With it off, it was outputting something like 23mA.
I dont know how many LEDs it has, it does not say on the datasheet (which is just a single sheet, or do I have the actual full datasheet?!)
I will buy a heatsink for the 7805.
My method of placing series resistance was to connect a potentiometer between the -ve terminal (ground) of the LCD, then I turned it (reduced the resistance) to get the LCD just ON (quite dim). But at decent backlight level, which is when it looks its best, the 7805 becomes really hot.
I think I just need a heatsink to dissipate all that heat.