"Big Boy" is absolutely correct regarding EL backlights.
Your module comes in three different versions - EL backlight, LED backlight, or no backlight. You need to find out which type you have, then power the backlight accordingly.
An LED backlight will only require a current limiting resistor in series with the DC power supply.
An EL backlight will require some higher AC voltage; HOWEVER, you should not make any assumptions about what voltage is required. EL backlights can require any voltage from tens of volts to several hundred volts depending on the type, thickness, and size. You will need data from the manufacturer regarding the required voltage and recommended frequency.
You may want to call Jaycar or Electus Distribution to see what they can tell you about the product.