How to check which parts of a dead MacBook Pro 17" Mid 2010 work
That will depend on what kind of analytical/measurement tools do you have? Oscilloscope, Volt meter ect...

Since it does not start at all and Battery is shut it could be simply the power input module. Open it and check if power is coming in and the fuse is not blown.

Is the display working: try the Target Display Mode.

Here is some how to:

http://forums.macrumors.com/showthread.php?t=1122338

Check your DCIN board.