Try putting the bad drive in the fridge for about 15 minutes. Sometimes it's a thermal expansion problem on the board or in a chip and you can get a few working minutes with the drive to copy files off. If that doesn't work, try the freezer. If that doesn't work, try some gentle heat with a hair dryer. If none of that works, you're back to the board swap or a professional recovery service. If the fridge/freezer thing works, using a USB interface on the drive will buy you some more up time, as you don't waste "cool" time while the machine boots up before you start pulling files off.
I agree about the freezer suggestion but you left out one VERY IMPORTANT precaution! The cold metal being re-exposed to a warmer environment with higher humidity when removed from the freezer will result in condensation on all external AND internal surfaces! The best way to do this safely is to put your internal drive into an external HD enclosure, connect the USB and power cord, then double bag it and tape the ends. You have to make the bag air tight but leave your cables hanging out. This way the drive is cooled with a fixed amount of air (and thus water particulate) surrounding it. As it cools a very small amount of moisture will be locked in the bag but not enough to be of concern. When it is removed from the freezer, this small amount of moisture is all it can/will be exposed to and further moisture from the surrounding air will be isolated. Once the drive has stabilized to ambient temperature it can be safely removed.
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