Hello
This absolutely sounds like the powersupply on the lcd screen is faulty, very common its the electrolyte capacitors that cost around 1 dollar each, and you can replace these yourself, with same value and voltage capacitors.
Another possible cause is the capacitors on the mainboard of the display, but it sounds like the mainboard is OK since you get it to start up sometimes, it seems like its not geting proper power from the powersupply.
Here is a picture of two GOOD capacitors
http://www.capacitorlab.com/capacitor-types-electrolytic/radial-capacitor-polarity.jpgOn a bad cap the surface, top, will not be flat, it might bulge even just a little bit and its difficult to notice.
You will probably get best and lasting results by replacing all the bigger capacitors but not the huge one.
To just fix the problem, you need to replace the faulty caps.
That should be it.
How to take apart an lcd display?
Unscrew all screws in the display, check under all labels and soft objects, remove the stand and check for additional screws.
Once you have confirmed that all screws are out, the casing is held in place with plastic clips that you can release with either your fingernails or a plastic stick or a flat screw driver on the edges openings of the casing, on the sides of the display. you need to go around the whole display until it comes apart, after that its pretty simple.
let me know if you have any questions.