If you look at this video. Ira T explains he uses 2 LEDs at the base of the display, in series, and connected (via switch) to a resistor and then to the battery.
Then if you read Ni5ei's post. Right below the negative battery contact on the board is a terminal labeled "V+" This is the 9V from the battery but switched on/off by the rotary dial. If you'd taken your + for the LED's from there it wouldn't be possible to leave the backlight switched on when you power the unit off.?
You can find photos
from this thread here showing where to mount the LED's.
So turn the meter on, backlight comes on. Turn multimeter off backlight goes off. Possibly the simplest circuit you can use, aside from the possibility of getting specific voltage LED.
As to what LED's to use you can use anything that will fit. If you don't know set yourself a challenge to learn how to power them and change their brightness with a resistor.