Function key emulator for Dell inspiron 6400 on external keyboard

If you think about it you'll see the answer is almost certainly NO. You say the GPU is dead. What is going to drive the external monitor... it's the GPU. THe Fn Key combo just tells the GPU output circuits to shunt or split the GPU output signal, it has no effect on how its created.

This is probably a motherboard issue

Your symptoms strongly suggest an electronic problem in the motherboard, since you've noted that it always happens before Windows starts booting up. It's quite unlikely that a laptop AC adapter would do this as the battery should be able to keep the system running even if the adapter failed. If you can confirm that the battery charges, your AC adapter is most likely good. Note that the CPU and GPU are soldered onto the motherboard on this laptop.

A hard drive failure wouldn't cause these kinds of boot-up issues as the system should be able to get through POST, just not load the OS. Bad or improperly-seated memory wouldn't explain this problem, either, as this would manifest as frequent random system and application crashes or the system giving a beep code when powered on, respectively. It's incredibly unlikely that these parts would cause the system to not clear POST with nary a beep code.

The motherboard is almost never a user-serviceable part in any laptop, so you should send it back to Dell for repair. Be sure to back up your data—the best way to do this is to remove the hard drive from the computer, put it in an enclosure, and plug it into another computer for data transfer or imaging.