They think it is your bother board. Here are three , maybe four, suggestions.
A. Contact a local Technician you trust and ask if he can fix it for under $100.
B. Bay a new battery and see if that makes a difference.
C. Buy a used Dell laptop, same mode, from eBay. (Be prudent.)
Now before you do the above, try this:
Remove the ac adapter, both ends, unplug from the wall and the Dell.
Next Remove the battery. Set it aside for now.
Now plug in the AC adapter to the wall and the Dell.
Now try the power button again.
Does that help any?