A number of things can go wrong in a TV, but here is a list of its subsystems:
1. Power board. Complete failure of this - TV will not turn on. Might wanna check the fuse or inspect the power board for signs of damage (burn marks, melted components etc) Look for bulged/leaked capacitors. I heard a few times that the large 150uF/450V cap might take damage from a power surge. Even if some parts of the power supply are fine, there might be a problem - i.e. the portion feeding 24V to the inverters might have gone bad. Again, capacitors are primary suspects. This is really the repair you should try first, if you can handle a soldering iron. Remove the caps, make careful notes on what was where. Order new ones at places like Digikey or Jameco. Get the same capacitance, and make sure the cap will physically fit the board - lead spacing, can diameter and height. Get highest voltage available, and opt for quality brand (i.e. Panasonic) rated for 2000-3000 or more hours at 105C. You can order caps on Digikey to replace every cap on the power board for less than $20, including shipping, and I really recommend you try that first before going for more expensive options. Also, VERY carefully, using a magnifying glass, inspect all solder joints - Say, if solder holding a connector pin has cracked, it could result in bad/no connection.
2. Backlight. You got two inverter boards - master and slave - hooked to 24V on the power board, and feeding the CCFLs. If the screen is black, but sound is there - backlight is likely the problem. Note, however, that problem might actually lie with the power board not feeding proper voltage - or simply a cracked-off connector pin resulting in a bad connection. Now, you can take the inverters out and examine the transformers carefully. Sometimes the damage is clearly visible. You can aso try and test the primary secondary windings using a multimeter. If you observe nice, steady 24-25V on the input pins at the inverter,(black to red) and control pins also have voltage (black/orange and black/green), the problem may indeed be the inverter. The models for LT37HVS are 6632L-0191A(master) and 6632L-0192A(slave). Best I seen was $170 for the pair. Still, before ordering, make doubly sure it's not the power board - and replace the caps.
2A. Sometimes, a particular CCFL in the backlight may fail, causing a shadowed band across the screen. It is possible to replace them, but requires almost complete TV disassembly.
3. T-Con. Hidden fairly deep, pretty much underneath the power board, this is the LCD control. Likely failure symptom: have sound and blue screen, but no image. LT37HVS T-Con model is 6871L-0662A, about $40. Again, this is something to try if playing with the power board did not help.
4. Main board. If you have blue screen, and no image or sound, then main board might be the problem. This is the most electronically complex part of the TV, and not much to do here other than replace it. But I understand is that of all parts, it is the least likely to fail.