Can I repair the Back light or must I replace LCD?

The LED backlights rarely fail, so I suspect a faulty panel as well. I would try replacing the LCD first. If you still have issues, the problem is likely related to the LED driver or supporting components. You may be able to repair the board, but there's instances the board is beyond repair and will need to be replaced. The coils rarely fail, so look at this last.

You're going to need to find a schematic of the board. The problem is this is difficult to find because Apple takes this down, so you need to find it and preserve it if you manage to find one. In terms of board repair the first thing I'd check is to check the fuses, resistors and capacitors since these generally fail first. If all of these fail, then the LED driver IC is at fault.

In terms of testing the fuse the meter needs to be put into continuity mode. If the meter beeps or says SHRT (or something to that effect), the fuses are okay. The capacitors can be tested in capacitance mode, but this mode isn't accurate on many meters. Use it as a reference, but don't completely rely on the readings. Test the resistors in resistance mode; if you need a tutorial, refer to this guide if you need to see how to do this. If both the fuses and resistors check out, then you will need to locate and replace the LED driver.