Hi, welcome to the forum.
It would be handy to have as much diagnostic info as possible, eg you say the power supply is at fault, how did you come to that conclusion ?.
I know from experience that the backlight system can be fussy and if the backlight bulbs are worn out, then the backlight chip just shuts down and so no backlight
With the power supply, first let it sit disconnected from the mains for 5 mins to allow any capacitors to discharge. Not having a circuit is a pain, however if you can identify the power supply controller chip you can look up the data sheet for it which give you an idea of how it works. A check for faulty bulging caps is a good idea, they tend to fail after a long time in service, they dry out. Also look for anything that looks broken, burnt, any fuses blown ?.
Next being very careful about live voltages on the power supply ( the big storage caps can be up at 375V DC ) check power rails are there and not loaded up, eg 5V is within 10% ish. The most common rails used are 5V, 3V3, and 12V so look for them, Basically look for the output connector, there should be several power rails fed there.
Do this and then report back.
Cheers Ken