Try charging the battery first. Its the easy cheap fix.
You may need to but a small trickle charger to keep the battery in top shape in the future.

If that fails, check your transformer. Hook it up to 12 volts and run a srewdriver sideways across the springs -- does it spark? yes = fine no= replace.

Been running the same machine for 14 years never replaced a transformer yet!

Although, it is time for a new one.

*****************CAUTION*************
The smoke is evident you are tossing large amounts of raw unignited diesel in your burner tube. This equals WOOOMF(say this at the top of your lungs) if it lights with heavy amounts of smoke.

Be careful after charging the battery and trying to ignite the burner. All that raw diesel will ignite at once and flames will discharge. (keep body parts clear).

Let it run if this happens until it burns clear. As long as the flames are contained in the tube.
Keep a fire extinguisher handy!

TTYL
Ben
Super Clean Presure Washing Service
Memphis/Atlanta