If the fan won't run at all, replace it - likewise with the backlight (or the inverter;  either could be faulty).

The fan is working now. I have also hooked an external monitor and that works fine. when I switch from external back to the main screen, that screen is fine for a few seconds then the back light goes out

Again, that could be a bad backlight or inverter- contact an LCD repair shop. Either will run $100 or so to fix.

My Inspiron 8200 has the same backlight problem. I took the screen apart and noticed that the orange-gold lead going into the inverter (bottom left) was loose.  Indeed, the lead does not have a plug on the end - it just slides up into the contact housing on the inverter (??).  I tried taping it in, but, no joy, I fear - the backlight is still going out after a few seconds or minutes.

So, I purchsed a "screen renewal" package from eBay, which was just a cracked screen and included and inverter and backlight.  I removed both (ugh).  I then removed the screen on my laptop and plugged the extra backlight into the existing inverter.  It lit up and stays lit (see pic). I now have the unenviable task of replacing the old backlight, which makes replacing the fan on this unit look like child's play.  But it is therapeutic.

update: all fixed after 90 minutes. I have extra screws leftover!