2010 FE390 Won't Start

sounds like a tough one. Going to put my money on electrical or clogged fuel injector/filter. There maybe an inline filter in your bike in the hose just before the injector. If you do, take it out. You can't see it, it's in the hose. Look around on the forum to see if your bike applies. Husaberg had a filter that was too fine(10 microns). I think on my 2011 they did away with it and then put a 20 micron filter back in the 2012/13's. The 09's had the fine one and some of the 2010's as well.

Check your terminals first, sometimes the bike will crank but not start just because of this...

The connector in front of the throttle body is plugged into the injector. The lock is on the top side and it can be fiddly to reach but possible.

Regarding removal of the injector the easiest, but longest way, is to remove the airbox and loosen the throttle body. It's possible to use flexible tools and unscrew the 2 screws that holds injector in place.
I would recommend though that to do it the long way the first time. Remove the airbox and loosen the throttle body and push it out on to the right side. When you done that you can unscrew the two phillips screws that holds down the injector.

A tip is to use the two allen key bolts that secure the plastic cover on the right side of the throttle body on the injector, and the phillips X bolts on the cover. That way you can use an allen key next time you want to check your injector. Then all you have to do is to use the round tip of a normal 4mm allen key, and/or reach in with the allen key from the right side. Then you save time leaving the airbox and throttle body in place.

Anyroad... When you have removed the injector you can open it by connecting a 9V battery to it. That will open it fully and you can blow air through it to, hopefully, remove dirt from it's filter. You should blow the air through the nozzle so any dirt is pushed back the way it came instead of into the injector.
You can also drench it in an injector cleaner fluid and blow it clean with air a couple of times. If you have access to an ultra sonic cleaner you can clean it there.

You can test the result by connecting the hose and injector to the tank and push the start button. Then you can watch the spray pattern of the fuel (point it into a bucket or something..). Do it before you start cleaning and after to see if you see a change. You can either connect the bikes plug into it to see the pattern when it starts, or hook up the 9V battery to see the pattern when it's fully open.

If it doesn't help, and your sure it's the injector, you have to take it to a professional cleaner or buy a new one.

Personally I have two injectors. I clean the one I use every 15 h with compressed air and every 45 h in an ultra sonic cleaner. The day I notice a performance difference between the cleaned injector and the new I know it's time to change it out.