I had one acting just like that - the left half died and the right half was working. Yours may be dead of other causes, but still this may give you some ideas.
With only half not working, I figured it was some loose connection. I tried pressing hard and/or flexing it a bit to force the contacts close - it worked for a while. I narrowed down to one of the two black globs (IC encased inside).
Mechanical pressure (rubber band) worked a while. Eventually, it totally died. I replaced mine. Just a few bucks. Mine had an I2C adapter which is still working on the replacement 2004 LCD.