We need some photos of your project, please.

This is important -- can be very helpful. And tell us about the CPU you're using -- was it purchased new from a reputable dealer, or is its origin somewhat doubtful?
Also, do you have a voltmeter or DMM available? (What about other gadgets, such as a 'scope or logic probe?) I suggest you use the meter to measure between Gnd and pin 40 (Reset). Please verify that the voltage is normally 5V, and that it goes to 0 when you hold the button down.
When I try to turn the computer on it just turns on the LCD backlight and I need to plug it in and out a lot of times so it can show the text on the screen.
Uhhh... to be clear, what is it you're plugging in and out? The LCD? The power cord for your PSU? Something else, perhaps?
Edit: is that resistor in your diagram really 220 ohm -- not 2200 ohm, perhaps?
-- Jeff