How to fix this constant current drive circuit?
There appears to be a couple of problems with your design:


D5 needs to be a Schottky diode (and as noted in another answer needs to be able to withstand the string voltage). I doubt that D1 is reverse avalanching since you don't report excessive heat, but you should select a more appropriate diode. 
The chip draws pulses of current up to 750mA, but with L1 being only 10uH you may well be putting the chip into overcurrent shutdown. When this happens the soft start is initiated so it is likely this what is preventing you achieving the right string voltage.


Why did you select an inductor half of the value recommended in the datasheet? See page 8.
Are you aware that this changes the peak current radically?