If you've installed a known working OEM fan and still see the error, the issue is with the fan controller.  Whether it's worth the cost to solve that problem (i.e., replacing the system board) is a question you'll have to consider on a system that's as old as this one. 
Hi, I hoping someone may have an answer for this one...  My XPS 15 9560 was running hot, so I opened and cleaned the fans. This fixed the temperature problem, but my CPU fan now is running at full speed rpm regardless of the temp. Tried running the diagnostics (boot up) and the fan failed the test. The test targets 2500rpm and the fan runs at 5800 rpm.  I figured it was a bad connection in the fan, so I replaced the fan with a new one, but am still having the same problem. It fails the diagnostic test, "The processor fan failed to respond correctly" (see image).  Any suggestions are greatly appreciated 
XPS 15 9560, replaced fan, fails diag test, runs at full speed