How do I convert rainfall (kg m-2 day-1) to rainfall (mm)?

I need to convert those units exactly as they are - thanks

You can convert Kg per square meter to mm by using the fact that 1 Kg of water occupies a volume of 1,000 cm^3. If you have 1000 cm^3 of water spread over 1 square meter (which is 100cm x 100cm = 10^4 cm^2) it would would be 0.1 cm high, or 1 mm. Hence 1 Kg/m^2 of rainfall is equaivalent to 1 mm.

Note that there is no "per day" here; if you want to talk ion terms of rainfall rates then 1Kg/(m^2-day) = 1 mm/day.