Counting the number of CPU cycles (and by extension the time) it takes to execute each instruction should help you understand how quickly this happens, so here I'll add a comment to each instruction saying how long it takes to run up to that point (assuming our Z80 is running at 6 MHz).
Code: OUT (10), A ; 11 cycles, 1.833 microseconds
CALL $000B ; unspecified number of cycles, 11.833 microseconds (10us delay for LCD)
LD A, $03 ; 7 cycles, 13 microseconds
OUT (10), A ; 11 cycles, 14.833 microseconds
CALL $000B ; another 10us delay, 24.833 microseconds
So the entire sequence to turn off the LCD and then turn it back on executes in about 25 microseconds; you could do this around 4000 times in the time it takes you to blink your eyes; that's much faster than you can see.
Calling _LCD_BUSY_QUICK ($000B) a few times isn't a terrible way to wait, but at 10 microseconds per call even calling it 10 times only delays for 100 microseconds which is still much too fast to see. Waiting a total of 100ms might be long enough to see, but it would probably be easiest to wait for any key to be pressed before turning it back on so the program will wait as long as you want:Code: ld a, $80
out (1), a
nop
nop
waitkey:
in a, (1)
cp $ff
jr nz, waitkey
(I would normally just loop on the result of GetCSC for this, but since you disabled interrupts the normal key input routines aren't available and we need to directly access the keypad.) 
