Unfortunately, having the computer start to work when you press down on a chip tends to indicate that some of the solder joints between the chip and the motherboard have failed and no longer provide an adequate electrical connection for the pins affected. This is especially true of Ball Grid Array (BGA) chip packages like the CPU and GPU here.
You may be able to get a better picture of what's going on if you remove the heatsink and power on the computer. Press down on the CPU, then the GPU and see which one causes the computer to start up. Naturally you will only want to leave it powered on for as short a time as possible when running without a heatsink; once it's going it'll heat up fairly quickly. Most likely it'll shut itself down before any damage occurs, but I personally wouldn't take that chance.
I'd say give that a try and let us know what happens. if pressing on one of them makes it work, then you've at least isolated the problem; at that point it will take someone who has expertise with BGA components to be able to remove the chip, reball the solder pads and reinstall the chip to fix a bad solder connection.
If neither the CPU or GPU fires it up, then we'll go back to the drawing board and try and figure out what else could be going on.
Good luck; let us know what you find!