It does sound like a power issue. I would try replacing the easiest thing first, which is the power button board that sits just behind the power button. Sellers on ebay sell them for under $10.00. They are not terribly hard to install. Remove the battery and unplug the power. Undo the screws from the bottom of the laptop. You just have to remove the palm rest, the speaker/keyboard bezel, the keyboard, and the screen bezel. Loosen up the right side hinge and you will see it. One screw and a tab hold it in place. The wires are extremely fragile and break away from the power button board really easily. Bless Dell's pea-pickin' heart. They designed the power button to be right where the hinge is, where the wire gets the most movement. When you replace it, you want to make sure that the wire is taught from the connector end and has lots of slack at the hinge end.

The power button board is bad. That is a small, PC board behind the power button that connects to the motherboard. I have had to replace several of these. They are small, surface-mount boards with three wires on them that come loose with the slightest tug. Unfortunately, on my Studio 1558, the power button is up by the right hinge, which when opened and closed, flexes the wire for the power button board. You have to remove the top portion of the laptop (the palm rest bezel, the keyboard/speaker bezel, and keyboard itself) along with the screen bezel. Then you have to loosen the hinge screws on the right to have the hinge move freely, so you can get to the power button board. One screw holds it in place plus a tab that goes into a plastic slot on the power button frame itself. Once you undo that, follow the cable and unhook it from the board. I got a new power button board from an eBay seller for under $10.00 and installed it. When installing, hook the power button board back in behind the power button,...