If I'm understanding you correctly...... a small cap came off the motor, (near the top of the router), the cap is broken and the spring is burned. What happen to the brush? Have you taken the other side off to see what it looks like?
You did not say what make of router you have but if nothing is wrong with the brush and you have not worn them down to the quick then most likely the vibration just caused the cap to come loose. In that case a simple call to the manufacture to order the broken part should be about all that is needed.
As far as what you were doing when this happen...... I might be wrong but that cut sounds a bit heavy..... but that is not what caused this to happen. You should check your brushes every (depends on how much you use it) couple of months and when you do this make sure the caps are on tight and in your case maybe a little more often since this happen.
If the brushes wear to far you can score the armature and then it will need to be turned or replaced.... this calls for a repair shop or a handy person
I live in Aurora IL and while I don't do repairs to anyone elses tools I do my own so I can't direct you anywhere, sorry.